JSR+matprod.bib

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@ARTICLE{AGGM:Automatica22,
 author        = "Aazan, Georges and Girard, Antoine and Greco, Luca and
                  Mason, Paolo",
 title         = "Stability of shuffled switched linear systems: {A} joint
                  spectral radius approach",
 journal       = "Automatica J. IFAC",
 fjournal      = "Automatica. A Journal of IFAC, the International Federation
                  of Automatic Control",
 year          = "2022",
 volume        = "143",
 pages         = "Paper No. 110434, 13",
 issn          = "0005-1098",
 mrclass       = "93C30 (93C55 93D05)",
 mrnumber      = "4444685",
 zblnumber     = "1497.93159",
 hal_id        = "hal-03257026",
 hal_version   = "v3",
 doi           = "10.1016/j.automatica.2022.110434",
 url           = "https://www.sciencedirect.com/science/article/pii/S0005109822002886",
 annote        = "A switching signal for a switched system is said to be
                  shuffled if all modes of the system are activated
                  infinitely often. In this paper, we develop tools to
                  analyze stability properties of discrete-time switched
                  linear systems driven by shuffled switching signals. We
                  introduce the new notion of shuffled joint spectral radius
                  (SJSR), which intuitively measures how much the state of
                  the system contracts each time the signal shuffles (i.e.
                  each time all modes have been activated). We show how this
                  notion relates to stability properties of the associated
                  switched systems. In particular, we show that some switched
                  systems that are unstable for arbitrary switching signals
                  can be stabilized by using switching signals that shuffle
                  sufficiently fast and that the SJSR allows us to derive an
                  expression of the minimal shuffling rate required to
                  stabilize the system. We then present several approaches to
                  compute lower and upper bounds of the SJSR using tools such
                  as the classical joint spectral radius, Lyapunov functions
                  and finite state automata. Several tightness results of the
                  bounds are established. Finally, numerical experiments are
                  presented to illustrate the main results of the paper.",
}

@INCOLLECTION{AGMG2024,
 author        = "Aazan, Georges and Girard, Antoine and Mason, Paolo and
                  Greco, Luca",
 editor        = "Postoyan, Romain and Frasca, Paolo and Panteley, Elena and
                  Zaccarian, Luca",
 title         = "A Joint Spectral Radius for {$\omega$}-Regular
                  Language-Driven Switched Linear Systems",
 booktitle     = "Hybrid and Networked Dynamical Systems: Modeling, Analysis
                  and Control",
 year          = "2024",
 publisher     = "Springer Nature Switzerland",
 address       = "Cham",
 pages         = "161--178",
 isbn          = "978-3-031-49555-7",
 doi           = "10.1007/978-3-031-49555-7_7",
 url           = "https://link.springer.com/chapter/10.1007/978-3-031-49555-7_7",
 annote        = "In this chapter, we introduce some tools to analyze
                  stability properties of discrete-time switched linear
                  systems driven by switching signals belonging to a given
                  $\omega$-regular language. More precisely, we assume that
                  the switching signals are generated by a deterministic
                  B{\"u}chi automaton whose alphabet coincides with the set
                  of modes of the switched system. We present the notion of
                  $\omega$-regular joint spectral radius ($\omega$-RJSR),
                  which intuitively describes the contraction of the state
                  when the run associated with the switching signal visits an
                  accepting state of the automaton. Then, we show how this
                  quantity is related to the stability properties of such
                  systems. Specifically, we show how this notion can
                  characterize a class of stabilizing switching signals for a
                  switched system that is unstable for arbitrary switching.
                  Though the introduced quantity is hard to compute, we
                  present some methods to approximate it using Lyapunov and
                  automata-theoretic techniques. More precisely, we show how
                  upper bounds can be computed by solving a convex
                  optimization problem. To validate the results of our work,
                  we finally show a numerical example related to the
                  synchronization of oscillators over a communication
                  network.",
}

@PHDTHESIS{Ahmadi08,
 author        = "Ahmadi, Amir Ali",
 authauthor    = "Ahmadi, Amir",
 title         = "Non-monotonic {L}yapunov functions for stability of
                  nonlinear and switched systems: theory and computation",
 school        = "Dept. of Electrical Engineering and Computer Science",
 address       = "Massachusetts Institute of Technology",
 year          = "2008",
 url           = "https://dspace.mit.edu/handle/1721.1/44206",
 annote        = "Lyapunov's direct method, which is based on the existence of
                  a scalar function of the state that decreases monotonically
                  along trajectories, still serves as the primary tool for
                  establishing stability of nonlinear systems. Since the main
                  challenge in stability analysis based on Lyapunov theory is
                  always to find a suitable Lyapunov function, weakening the
                  requirements of the Lyapunov function is of great interest.
                  In this thesis, we relax the monotonicity requirement of
                  Lyapunov's theorem to enlarge the class of functions that
                  can provide certificates of stability. Both the discrete
                  time case and the continuous time case are covered.
                  Throughout the thesis, special attention is given to
                  techniques from convex optimization that allow for
                  computationally tractable ways of searching for Lyapunov
                  functions. Our theoretical contributions are therefore
                  amenable to convex programming formulations. In the
                  discrete time case, we propose two new sufficient
                  conditions for global asymptotic stability that allow the
                  Lyapunov functions to increase locally, but guarantee an
                  average decrease every few steps. Our first condition is
                  nonconvex, but allows an intuitive interpretation. The
                  second condition, which includes the first one as a special
                  case, is convex and can be cast as a semidefinite program.
                  We show that when non-monotonic Lyapunov functions exist,
                  one can construct a more complicated function that
                  decreases monotonically. We demonstrate the strength of our
                  methodology over standard Lyapunov theory through examples
                  from three different classes of dynamical systems. First,
                  we consider polynomial dynamics where we utilize techniques
                  from sum-of-squares programming. Second, analysis of
                  piecewise and systems is performed. Here, connections to
                  the method of piecewise quadratic Lyapunov functions are
                  made.(cont.) Finally, we examine systems with arbitrary
                  switching between a finite set of matrices. It will be
                  shown that tighter bounds on the joint spectral radius can
                  be obtained using our technique. In continuous time, we
                  present conditions invoking higher derivatives of Lyapunov
                  functions that allow the Lyapunov function to increase but
                  bound the rate at which the increase can happen. Here, we
                  build on previous work by Butz that provides a nonconvex
                  sufficient condition for asymptotic stability using the
                  first three derivatives of Lyapunov functions. We give a
                  convex condition for asymptotic stability that includes the
                  condition by Butz as a special case. Once again, we draw
                  the connection to standard Lyapunov functions. An example
                  of a polynomial vector field is given to show the potential
                  advantages of using higher order derivatives over standard
                  Lyapunov theory. We also discuss a theorem by Yorke that
                  imposes minor conditions on the first and second
                  derivatives to reject existence of periodic orbits, limit
                  cycles, or chaotic attractors. We give some simple convex
                  conditions that imply the requirement by Yorke and we
                  compare them with those given in another earlier work.
                  Before presenting our main contributions, we review some
                  aspects of convex programming with more emphasis on
                  semidefinite programming. We explain in detail how the
                  method of sum of squares decomposition can be used to
                  efficiently search for polynomial Lyapunov functions.",
}

@INPROCEEDINGS{AhmJun:CDC13,
 author        = "Ahmadi, Amir Ali and Jungers, Rapha{\"e}l M.",
 authauthor    = "Ahmadi, Amir and Rapha{\"e}l Jungers",
 title         = "Switched stability of nonlinear systems via {SOS}-convex
                  {L}yapunov functions and semidefinite programming",
 booktitle     = "Proceedings of the 52nd IEEE Annual Conference on Decision
                  and Control (CDC)",
 year          = "2013",
 pages         = "727--732",
 doi           = "10.1109/CDC.2013.6759968",
 url           = "https://ieeexplore.ieee.org/document/6759968",
 annote        = "We introduce the concept of sos-convex Lyapunov functions
                  for stability analysis of discrete time switched systems.
                  These are polynomial Lyapunov functions that have an
                  algebraic certificate of convexity, and can be efficiently
                  found by semidefinite programming. We show that convex
                  polynomial Lyapunov functions are universal (i.e.,
                  necessary and sufficient) for stability analysis of
                  switched linear systems. On the other hand, we show via an
                  explicit example that the minimum degree of an sos-convex
                  Lyapunov function can be arbitrarily higher than a
                  (non-convex) polynomial Lyapunov function. (The proof is
                  omitted.) In the second part, we show that if the switched
                  system is defined as the convex hull of a finite number of
                  nonlinear functions, then existence of a non-convex common
                  Lyapunov function is not a sufficient condition for
                  switched stability, but existence of a convex common
                  Lyapunov function is. This shows the usefulness of the
                  computational machinery of sos-convex Lyapunov functions
                  which can be applied either directly to the switched
                  nonlinear system, or to its linearization, to provide proof
                  of local switched stability for the nonlinear system. An
                  example is given where no polynomial of degree less than 14
                  can provide an estimate to the region of attraction under
                  arbitrary switching.",
}

@ARTICLE{AhmJun14,
 author        = "Ahmadi, Amir Ali and Jungers, Rapha{\"e}l M.",
 authauthor    = "Ahmadi, Amir and Jungers, Rapha{\"e}l",
 title         = "On Complexity of {L}yapunov Functions for Switched Linear
                  Systems",
 journal       = "IFAC Proceedings Volumes",
 year          = "2014",
 volume        = "47",
 number        = "3",
 pages         = "5992--5997",
 issn          = "1474-6670",
 doi           = "10.3182/20140824-6-ZA-1003.02484",
 url           = "https://www.sciencedirect.com/science/article/pii/S1474667016425504",
 note          = "19th IFAC World Congress",
 annote        = "We show that for any positive integer $d$, there are
                  families of switched linear systems --- in fixed dimension
                  and defined by two matrices only --- that are stable under
                  arbitrary switching but do not admit (i) a polynomial
                  Lyapunov function of degree $\le d$, or (ii) a polytopic
                  Lyapunov function with $\le d$ facets, or (iii) a piecewise
                  quadratic Lyapunov function with $\le d$ pieces. This
                  implies that there cannot be an upper bound on the size of
                  the linear and semidefinite programs that search for such
                  stability certificates. Several constructive and
                  non-constructive arguments are presented which connect our
                  problem to known (and rather classical) results in the
                  literature regarding the finiteness conjecture,
                  undecidability, and non-algebraicity of the joint spectral
                  radius. In particular, we show that existence of a sum of
                  squares Lyapunov function implies the finiteness property
                  of the optimal product.",
}

@ARTICLE{AhmJng:NAHS16,
 author        = "Ahmadi, Amir Ali and Jungers, Rapha{\"e}l M.",
 authauthor    = "Ahmadi, Amir and Jungers, Rapha{\"e}l",
 title         = "Lower bounds on complexity of {L}yapunov functions for
                  switched linear systems",
 journal       = "Nonlinear Anal. Hybrid Syst.",
 fjournal      = "Nonlinear Analysis. Hybrid Systems",
 year          = "2016",
 volume        = "21",
 pages         = "118--129",
 eprinttype    = "arXiv",
 eprint        = "1504.03761",
 issn          = "1751-570X",
 mrclass       = "93D30 (90C90)",
 mrnumber      = "3500076",
 zblnumber     = "1382.93028",
 doi           = "10.1016/j.nahs.2016.01.003",
 url           = "https://www.sciencedirect.com/science/article/pii/S1751570X16000121",
 annote        = "We show that for any positive integer $d$, there are
                  families of switched linear systems --- in fixed dimension
                  and defined by two matrices only --- that are stable under
                  arbitrary switching but do not admit (i) a polynomial
                  Lyapunov function of degree $\le d$, or (ii) a polytopic
                  Lyapunov function with $\le d$ facets, or (iii) a piecewise
                  quadratic Lyapunov function with $\le d$ pieces. This
                  implies that there cannot be an upper bound on the size of
                  the linear and semidefinite programs that search for such
                  stability certificates. Several constructive and
                  non-constructive arguments are presented which connect our
                  problem to known (and rather classical) results in the
                  literature regarding the finiteness conjecture,
                  undecidability, and non-algebraicity of the joint spectral
                  radius. In particular, we show that existence of an
                  extremal piecewise algebraic Lyapunov function implies the
                  finiteness property of the optimal product, generalizing a
                  result of Lagarias and Wang. As a corollary, we prove that
                  the finiteness property holds for sets of matrices with an
                  extremal Lyapunov function belonging to some of the most
                  popular function classes in controls.",
}

@ARTICLE{AJPR:IFACSRCD12,
 author        = "Ahmadi, Amir Ali and Jungers, Raphael M. and Parrilo, Pablo
                  A. and Roozbehani, Mardavij",
 authauthor    = "Ahmadi, Amir and Jungers, Raphael and Parrilo, Pablo and
                  Roozbehani, Mardavij",
 title         = "When is a set of {LMI}s a sufficient condition for
                  stability?",
 journal       = "IFAC Proceedings Volumes",
 year          = "2012",
 volume        = "45",
 number        = "13",
 pages         = "313--318",
 eprinttype    = "arXiv",
 eprint        = "1201.3227",
 issn          = "1474-6670",
 doi           = "10.3182/20120620-3-DK-2025.00098",
 url           = "https://www.sciencedirect.com/science/article/pii/S1474667015377077",
 note          = "7th IFAC Symposium on Robust Control Design",
 annote        = "We study stability criteria for discrete time switching
                  systems. We investigate the structure of sets of LMIs that
                  are a sufficient condition for stability (i.e., such that
                  any switching system which satisfies these LMIs is stable).
                  We provide an exact characterization of these sets. As a
                  corollary, we show that it is PSPACE-complete to recognize
                  whether a particular set of LMIs implies the stability of a
                  switching system.",
}

@ARTICLE{AJPR:SIAMJCO14,
 author        = "Ahmadi, Amir Ali and Jungers, Rapha{\"e}l M. and Parrilo,
                  Pablo A. and Roozbehani, Mardavij",
 authauthor    = "Ahmadi, Amir and Jungers, Rapha{\"e}l and Parrilo, Pablo and
                  Roozbehani, Mardavij",
 title         = "Joint spectral radius and path-complete graph {L}yapunov
                  functions",
 journal       = "SIAM J. Control Optim.",
 fjournal      = "SIAM Journal on Control and Optimization",
 year          = "2014",
 volume        = "52",
 number        = "1",
 pages         = "687--717",
 eprinttype    = "arXiv",
 eprint        = "1111.3427",
 issn          = "0363-0129",
 mrclass       = "93D30 (37C75)",
 mrnumber      = "3168608",
 mrreviewer    = "Bryan E. Cain",
 zblnumber     = "1292.93093",
 doi           = "10.1137/110855272",
 url           = "https://epubs.siam.org/doi/10.1137/110855272",
 annote        = "We introduce the framework of path-complete graph Lyapunov
                  functions for approximation of the joint spectral radius.
                  The approach is based on the analysis of the underlying
                  switched system via inequalities imposed among multiple
                  Lyapunov functions associated to a labeled directed graph.
                  Inspired by concepts in automata theory and symbolic
                  dynamics, we define a class of graphs called path-complete
                  graphs, and show that any such graph gives rise to a method
                  for proving stability of the switched system. This enables
                  us to derive several asymptotically tight hierarchies of
                  semidefinite programming relaxations that unify and
                  generalize many existing techniques such as common
                  quadratic, common sum of squares, and
                  maximum/minimum-of-quadratics Lyapunov functions. We
                  compare the quality of approximation obtained by certain
                  classes of path-complete graphs including a family of dual
                  graphs and all path-complete graphs with two nodes on an
                  alphabet of two matrices. We provide approximation
                  guarantees for several families of path-complete graphs,
                  such as the De Bruijn graphs, establishing as a byproduct a
                  constructive converse Lyapunov theorem for
                  maximum/minimum-of-quadratics Lyapunov functions.",
}

@INPROCEEDINGS{AJPR:HCSS11,
 author        = "Ahmadi, Amir Ali and Jungers, Rapha{\"e}l and Parrilo, Pablo
                  A. and Roozbehani, Mardavij",
 authauthor    = "Ahmadi, Amir and Jungers, Rapha{\"e}l and Parrilo, Pablo and
                  Roozbehani, Mardavij",
 title         = "Analysis of the joint spectral radius via {L}yapunov
                  functions on path-complete graphs",
 booktitle     = "Proceedings of the 14th international conference on Hybrid
                  systems: computation and control (HSCC'11)",
 publisher     = "ACM",
 address       = "New York, NY, USA",
 year          = "2011",
 pages         = "13--22",
 mrnumber      = "3207287",
 zblnumber     = "1364.93376",
 doi           = "10.1145/1967701.1967706",
 url           = "https://dl.acm.org/doi/10.1145/1967701.1967706",
 annote        = "We study the problem of approximating the joint spectral
                  radius (JSR) of a finite set of matrices. Our approach is
                  based on the analysis of the underlying switched linear
                  system via inequalities imposed between multiple Lyapunov
                  functions associated to a labeled directed graph. Inspired
                  by concepts in automata theory and symbolic dynamics, we
                  define a class of graphs called path-complete graphs, and
                  show that any such graph gives rise to a method for proving
                  stability of the switched system. This enables us to derive
                  several asymptotically tight hierarchies of semidefinite
                  programming relaxations that unify and generalize many
                  existing techniques such as common quadratic, common sum of
                  squares, maximum/minimum-of-quadratics Lyapunov functions.
                  We characterize all path-complete graphs consisting of two
                  nodes on an alphabet of two matrices and compare their
                  performance. For the general case of any set of $n\times n$
                  matrices we propose semidefinite programs of modest size
                  that approximate the JSR within a multiplicative factor of
                  $1/\sqrt[4]{n}$ of the true value. We establish a notion of
                  duality among path-complete graphs and a constructive
                  converse Lyapunov theorem for maximum/minimum-of-quadratics
                  Lyapunov functions.",
}

@INPROCEEDINGS{AhmPar:CDC05,
 author        = "Ahmadi, Amir Ali and Parrilo, Pablo A.",
 authauthor    = "Ahmadi, Amir and Parrilo, Pablo",
 title         = "Non-monotonic {L}yapunov functions for stability of discrete
                  time nonlinear and switched systems",
 booktitle     = "Proceedings of the 47th IEEE Conference on Decision and
                  Control (CDC)",
 year          = "2008",
 pages         = "614--621",
 doi           = "10.1109/CDC.2008.4739402",
 url           = "https://ieeexplore.ieee.org/document/4739402",
 annote        = "We relax the monotonicity requirement of Lyapunov's theorem
                  to enlarge the class of functions that can provide
                  certificates of stability. To this end, we propose two new
                  sufficient conditions for global asymptotic stability that
                  allow the Lyapunov functions to increase locally, but
                  guarantee an average decrease every few steps. Our first
                  condition is non-convex, but allows an intuitive
                  interpretation. The second condition, which includes the
                  first one as a special case, is convex and can be cast as a
                  semidefinite program. We show that when non-monotonic
                  Lyapunov functions exist, one can construct a more
                  complicated function that decreases monotonically. We
                  demonstrate the strength of our methodology over standard
                  Lyapunov theory through examples from three different
                  classes of dynamical systems. First, we consider polynomial
                  dynamics where we utilize techniques from sum-of-squares
                  programming. Second, analysis of piecewise affine systems
                  is performed. Here, connections to the method of piecewise
                  quadratic Lyapunov functions are made. Finally, we examine
                  systems with arbitrary switching between a finite set of
                  matrices. It will be shown that tighter bounds on the joint
                  spectral radius can be obtained using our technique.",
}

@INPROCEEDINGS{AhmPar:CDC12,
 author        = "Ahmadi, Amir Ali and Parrilo, Pablo A.",
 authauthor    = "Ahmadi, Amir and Parrilo, Pablo",
 title         = "Joint Spectral Radius of Rank One Matrices and the Maximum
                  Cycle Mean Problem",
 booktitle     = "Proceedings of the 51st Annual Conference on Decision and
                  Control (CDC), 10-13 Dec.",
 organization  = "IEEE",
 year          = "2012",
 pages         = "731--733",
 doi           = "10.1109/CDC.2012.6425992",
 url           = "https://ieeexplore.ieee.org/document/1582513",
 annote        = "We prove several exact results on approximability of joint
                  spectral radius by matrix norms induced by Euclidean norms.
                  We point out, perhaps for the first time in this context, a
                  difference between complex and real cases. New connections
                  of joint spectral radius to convex geometry and
                  combinatorics are established. Several open problems are
                  posed.",
}

@INPROCEEDINGS{ABMW:MTNS12,
 author        = "Ait Rami, Mustafa and Bokharaie, Vahid S. and Mason, Oliver
                  and Wirth, Fabian R.",
 authauthor    = "Ait Rami, Mustafa and Bokharaie, Vahid and Mason, Oliver and
                  Wirth, Fabian",
 title         = "Extremal norms for positive linear inclusions",
 booktitle     = "20th International Symposium on Mathematical Theory of
                  Networks and Systems, MTNS2012, 9--13 July",
 organization  = "The University of Melburne",
 address       = "Melbourne",
 year          = "2012",
 url           = "https://citeseerx.ist.psu.edu/document?type=pdf&doi=224a9b6ba92e969309775ff068fd1ad522692483",
 annote        = "We consider the joint spectral radius of sets of matrices
                  for discrete or continuous positive linear inclusions and
                  study associated extremal norms. We show that under a
                  matrix-theoretic notion of irreducibility there exist
                  absolute extremal norms. This property is used to extend
                  regularity results for the joint spectral radius. In
                  particular, we see that in the case of positive systems
                  irreducibility in the sense of nonnegative matrices, which
                  is weaker than the usual representation theoretic concept,
                  is sufficient for local Lipschitz properties of the joint
                  spectral radius.",
}

@ARTICLE{Alpin:MZ10,
 author        = "Al'pin, {\relax{}Yu}. A.",
 authauthor    = "Al'pin, {\relax{}Yu}.",
 title         = "Bounds for the joint spectral radii of a set of nonnegative
                  matrices",
 journal       = "Math. Notes",
 fjournal      = "Mathematical Notes",
 year          = "2010",
 volume        = "87",
 number        = "1",
 pages         = "12--14",
 issn          = "0001-4346",
 mrclass       = "15B48 (15A42)",
 mrnumber      = "2730378 (2011j:15054)",
 mrreviewer    = "Mihail Voicu",
 zblnumber     = "1202.15022",
 zblreviewer   = "Ludwig Elsner (Bielefeld)",
 doi           = "10.1134/S0001434610010025",
 url           = "https://link.springer.com/article/10.1134/S0001434610010025",
 annote        = "For a finite set $\Sigma$ of nonnegative matrices of order
                  $n$ the upper and lower joint spectral radius is defined.
                  Bounds for these magnitudes are given in terms of the
                  following matrix $S=(s_{ij})$, where $s_{ij}
                  =\max\{\sum_{k} a_{ik} : A\in\Sigma, a_{ij} >0\}$ and in
                  terms of the similarly defined matrix $S^{-}$. It is shown
                  that the upper joint spectral radius is bounded by the max
                  spectral radius of S and that a similar result holds for
                  the lower joint spectral radius. This generalizes results
                  for the case of one matrix. In the proof methods of max
                  algebra are used.",
}

@ARTICLE{ASU:I11,
 author        = "Allen, Jeffrey and Seeger, Benjamin and Unger, Deborah",
 title         = "On the size of the resonant set for the products of
                  {$2\times 2$} matrices",
 journal       = "Involve",
 fjournal      = "Involve. A Journal of Mathematics",
 year          = "2011",
 volume        = "4",
 number        = "2",
 pages         = "157--166",
 eprinttype    = "arXiv",
 eprint        = "1106.2903",
 issn          = "1944-4176",
 mrclass       = "37H15 (37C85 37H05)",
 mrnumber      = "2876196",
 zblnumber     = "1233.37029",
 doi           = "10.2140/involve.2011.4.157",
 url           = "https://msp.org/involve/2011/4-2/p05.xhtml",
 annote        = "For $\theta\in[0,2\pi)$, consider the rotation matrix
                  $R_\theta$ and \[h=\left(\begin{array}{cc}\lambda & 0\\ 0 &
                  0\end{array}\right),\quad\lambda>1.\] Let $W_n(\theta)$
                  denote the product of $m$ $R_{\theta}$'s and $n$ $h$'s with
                  the condition $m\leq[\epsilon n]$ ($0<\epsilon< 1$). We
                  analyze the measure of the set of $\theta$ for which
                  $\|W_n( \theta)\|\geq\lambda^{\delta n}$ ($0<\delta< 1$).
                  This can be regarded as a model problem for the so-called
                  Bochi--Fayad conjecture.",
}

@INPROCEEDINGS{AltPar:CDC19,
 author        = "Altschuler, Jason M. and Parrilo, Pablo A.",
 authauthor    = "Jason Altschuler and Parrilo, Pablo",
 title         = "Lyapunov exponent of rank-one matrices: ergodic formula and
                  inapproximability of the optimal distribution",
 booktitle     = "2019 IEEE 58th Conference on Decision and Control (CDC),
                  11-13 Dec.",
 publisher     = "IEEE",
 address       = "Nice, France, France",
 year          = "2019",
 pages         = "4439--4445",
 eprinttype    = "arXiv",
 eprint        = "1905.07531",
 doi           = "10.1109/CDC40024.2019.9029462",
 url           = "https://ieeexplore.ieee.org/document/9029462",
 annote        = "The \emph{Lyapunov exponent} corresponding to a set of
                  square matrices $\mathcal{A} = \{A_1, \dots, A_n \}$ and a
                  probability distribution $p$ over $\{1, \dots, n\}$ is
                  $\lambda(\mathcal{A}, p) := \lim_{k \to \infty} \frac{1}{k}
                  \,\mathbb{E} \log \lVert{A_{\sigma_k} \cdots
                  A_{\sigma_2}A_{\sigma_1}}\rVert$, where $\sigma_i$ are
                  i.i.d. according to $p$. This quantity is of fundamental
                  importance to control theory since it determines the
                  asymptotic convergence rate $e^{\lambda(\mathcal{A}, p)}$
                  of the stochastic linear dynamical system $x_{k+1} =
                  A_{\sigma_k} x_k$. This paper investigates the following
                  ``design problem'': given $\mathcal{A}$, compute the
                  distribution $p$ minimizing $\lambda(\mathcal{A}, p)$. Our
                  main result is that it is NP-hard to decide whether there
                  exists a distribution $p$ for which $\lambda(\mathcal{A},
                  p) < 0$, i.e. it is NP-hard to decide whether this
                  dynamical system can be stabilized.\par This hardness
                  result holds even in the ``simple'' case where
                  $\mathcal{A}$ contains only rank-one matrices. Somewhat
                  surprisingly, this is in stark contrast to the Joint
                  Spectral Radius -- the deterministic kindred of the
                  Lyapunov exponent -- for which the analogous optimization
                  problem for rank-one matrices is known to be exactly
                  computable in polynomial time.\par To prove this hardness
                  result, we first observe via Birkhoff's Ergodic Theorem
                  that the Lyapunov exponent of rank-one matrices admits a
                  simple formula and in fact is a quadratic form in~$p$.
                  Hardness of the design problem is shown through a reduction
                  from the Independent Set problem. Along the way, simple
                  examples are given illustrating that $p \mapsto
                  \lambda(\mathcal{A}, p)$ is neither convex nor concave in
                  general, and a connection is made to the fact that the
                  Martin distance on the $(1, n)$ Grassmannian is not a
                  metric. See.~\cite{AltPar:SIAMJCO20}.",
}

@ARTICLE{AltPar:SIAMJCO20,
 author        = "Altschuler, Jason M. and Parrilo, Pablo A.",
 authauthor    = "Jason Altschuler and Parrilo, Pablo",
 title         = "Lyapunov exponent of rank-one matrices: ergodic formula and
                  inapproximability of the optimal distribution",
 journal       = "SIAM J. Control Optim.",
 fjournal      = "SIAM Journal on Control and Optimization",
 year          = "2020",
 volume        = "58",
 number        = "1",
 pages         = "510--528",
 issn          = "0363-0129",
 mrclass       = "37M25 (68Q17)",
 mrnumber      = "4068319",
 zblnumber     = "1451.93408",
 doi           = "10.1137/19M1264072",
 url           = "https://epubs.siam.org/doi/10.1137/19M1264072",
 annote        = "The \emph{Lyapunov exponent} corresponding to a set of
                  square matrices $\mathcal{A} = \{A_1, \dots, A_n \}$ and a
                  probability distribution $p$ over $\{1, \dots, n\}$ is
                  $\lambda(\mathcal{A}, p) := \lim_{k \to \infty} \frac{1}{k}
                  \,\mathbb{E} \log \lVert{A_{\sigma_k} \cdots
                  A_{\sigma_2}A_{\sigma_1}}\rVert$, where $\sigma_i$ are
                  i.i.d. according to $p$. This quantity is of fundamental
                  importance to control theory since it determines the
                  asymptotic convergence rate $e^{\lambda(\mathcal{A}, p)}$
                  of the stochastic linear dynamical system $x_{k+1} =
                  A_{\sigma_k} x_k$. This paper investigates the following
                  ``design problem'': given $\mathcal{A}$, compute the
                  distribution $p$ minimizing $\lambda(\mathcal{A}, p)$. Our
                  main result is that it is NP-hard to decide whether there
                  exists a distribution $p$ for which $\lambda(\mathcal{A},
                  p) < 0$, i.e. it is NP-hard to decide whether this
                  dynamical system can be stabilized.\par This hardness
                  result holds even in the ``simple'' case where
                  $\mathcal{A}$ contains only rank-one matrices. Somewhat
                  surprisingly, this is in stark contrast to the Joint
                  Spectral Radius -- the deterministic kindred of the
                  Lyapunov exponent -- for which the analogous optimization
                  problem for rank-one matrices is known to be exactly
                  computable in polynomial time.\par To prove this hardness
                  result, we first observe via Birkhoff's Ergodic Theorem
                  that the Lyapunov exponent of rank-one matrices admits a
                  simple formula and in fact is a quadratic form in~$p$.
                  Hardness of the design problem is shown through a reduction
                  from the Independent Set problem. Along the way, simple
                  examples are given illustrating that $p \mapsto
                  \lambda(\mathcal{A}, p)$ is neither convex nor concave in
                  general. We conclude with extensions to continuous
                  distributions, exchangeable processes, Markov processes,
                  and stationary ergodic processes.
                  See.~\cite{AltPar:CDC19}.",
}

@ARTICLE{AndoShih:SIAM:98,
 author        = "Ando, Tsuyoshi and Shih, Mau-Hsiang",
 title         = "Simultaneous contractibility",
 journal       = "SIAM J. Matrix Anal. Appl.",
 fjournal      = "SIAM Journal on Matrix Analysis and Applications",
 year          = "1998",
 volume        = "19",
 number        = "2",
 pages         = "487--498 (electronic)",
 issn          = "0895-4798",
 mrclass       = "15A60 (47A30)",
 mrnumber      = "1614074 (99c:15046)",
 mrreviewer    = "Sivaram K. Narayan",
 zblnumber     = "0912.15033",
 zblreviewer   = "Alexey Alimov (Moskva)",
 doi           = "10.1137/S0895479897318812",
 url           = "https://epubs.siam.org/doi/10.1137/S0895479897318812",
 annote        = "Let $\mathcal{C}$ be a set of $n\times n$ complex matrices.
                  For $m = 1,2,\ldots$, $\mathcal{C}^{m}$ is the set of all
                  products of matrices in $\mathcal{C}$ of length $m$. Denote
                  by $\hat{r}(\mathcal{C})$ the joint spectral radius of
                  $\mathcal{C}$, that is, \[\hat{r}(\mathcal{C})\stackrel{\rm
                  def}{=}\limsup_{m\to\infty}[\sup_{A\in \mathcal{C}^m}\| A\|
                  ]^{\frac{1}{m}}.\] We call $\mathcal{C}$
                  \emph{simultaneously contractible} if there is an
                  invertible matrix $S$ such that \[\sup\{\| S^{-1}AS\|;\
                  A\in \mathcal{C}\}<1,\] where $\| \cdot\|$ is the spectral
                  norm. This paper is primarily devoted to determining the
                  optimal joint spectral radius range for simultaneous
                  contractibility of bounded sets of $n\times n$ complex
                  matrices, that is, the maximum subset $J$ of $[0,1)$ such
                  that if $\mathcal{C}$ is a bounded set of $n\times n$
                  complex matrices and $\hat{r}(\mathcal{C})\in J$, then
                  $\mathcal{C}$ is simultaneously contractible. The central
                  result proved in this paper is that this maximum subset is
                  $[0,\frac{1}{\sqrt{n}})$. Our method of proof is based on a
                  matrix-theoretic version of complex John's ellipsoid
                  theorem and the generalized Gelfand spectral radius
                  formula.",
}

@ARTICLE{AT:JMAA77,
 author        = "Anthonisse, Jac. M. and Tijms, Henk",
 authauthor    = "Anthonisse, Jac. and Tijms, Henk",
 title         = "Exponential convergence of products of stochastic matrices",
 journal       = "J. Math. Anal. Appl.",
 fjournal      = "Journal of Mathematical Analysis and Applications",
 year          = "1977",
 volume        = "59",
 number        = "2",
 pages         = "360--364",
 issn          = "0022-247x",
 mrclass       = "60J10 (15A51)",
 mrnumber      = "0443092 (56 \#1465)",
 mrreviewer    = "E. Seneta",
 zblnumber     = "0381.15011",
 doi           = "10.1016/0022-247X(77)90114-7",
 annote        = "This paper considers a finite set of stochastic matrices of
                  finite order. Conditions are given under which any product
                  of matrices from this set converges to a constant
                  stochastic matrix. Also, it is shown that the convergence
                  is exponentially fast.",
}

@BOOK{AKKK:92:e,
 author        = "Asarin, E. A. and Kozyakin, V. S. and Krasnosel'ski{\u\i},
                  M. A. and Kuznetsov, N. A.",
 authauthor    = "Asarin, E. and Kozyakin, Victor and Krasnosel'skii, Mark and
                  Kuznetsov, N.",
 title         = "Analiz ustoichivosti rassinkhronizovannykh diskretnykh
                  sistem",
 publisher     = "Nauka",
 address       = "Moscow",
 year          = "1992",
 pages         = "408",
 isbn          = "5-02-006946-9",
 mrclass       = "93D05 (39A11)",
 mrnumber      = "1693324",
 url           = "http://eqworld.ipmnet.ru/ru/library/books/AsarinKozyakinKrasnoselskijKuznecov1992ru.pdf",
 note          = "In Russian, English title: ``Stability analysis of
                  desynchronized (asynchronous) discrete-time systems''",
 annote        = "Control systems with asynchronous data exchange are
                  considered. The stability theory of such systems (for
                  various desynchronization classes) is presented. Some
                  results may be considered as a new chapter of the theory of
                  matrices. The correctness of the proposed models is also
                  investigated. Many open problems are set forth. For
                  engineers, mathematicians and control theorists. In
                  Russian.",
}

@ARTICLE{AKKK:MCM90,
 author        = "Asarin, E. A. and Krasnoselskii, M. A. and Kozyakin, V. S.
                  and Kuznetsov, N. A.",
 authauthor    = "Asarin, E. and Krasnosel'skii, Mark and Kozyakin, Victor and
                  Kuznetsov, N.",
 title         = "On modelling systems with non-synchronously operating
                  impulse elements",
 journal       = "Math. Comput. Model.",
 fjournal      = "Mathematical and Computer Modelling",
 year          = "1990",
 volume        = "14",
 number        = "C",
 pages         = "70--73",
 zblnumber     = "0744.93082",
 zblreviewer   = "E. Eitelberg (Durban)",
 doi           = "10.1016/0895-7177(90)90149-H",
 annote        = "Systems with impulse elements operating by some reason
                  non-synchronously are often used in control science,
                  technique, economics, ecology etc. To analyze such
                  desynchronized systems one needs new constructions. Some
                  results connected with new models and computer methods of
                  stability analysis are set forth. The methods of stability
                  analysis of synchronized systems are well developed. As a
                  computer experiment has demonstrated all the conceivable
                  effects of synchronized systems loosing, gaining or
                  conserving stability are possible. So desynchronized
                  systems require the development of special methods of
                  stability analysis. In some simple cases the conditions of
                  stability can be expressed by formulae. In other situations
                  numerical methods of stability analysis of desynchronized
                  systems are to be developed. Three situations of this kind
                  are discussed in detail. 1. Phase and frequency
                  desynchronized systems, i.e. linear systems with
                  periodically switched components are considered. 2. It is
                  discussed, what is to be done if a system is sensitive to
                  desynchronization (looses stability under
                  desynchronization). A method to do away with such a
                  sensitivity is proposed. 3. Stochastically desynchronized
                  systems are considered.",
}

@ARTICLE{ADFP:93,
 author        = "Asarin, E. and Diamond, Phil and Fomenko, I. and Pokrovskii,
                  A.",
 title         = "Chaotic phenomena in desynchronized systems and stability
                  analysis",
 journal       = "Comput. Math. Appl.",
 fjournal      = "Computers {\&} Mathematics with Applications. An
                  International Journal",
 year          = "1993",
 volume        = "25",
 number        = "1",
 pages         = "81--87",
 coden         = "CMAPDK",
 issn          = "0898-1221",
 mrclass       = "58F13 (58F03 58F10 93D05)",
 mrnumber      = "1192675 (93k:58144)",
 mrreviewer    = "Peter E. Kloeden",
 zblnumber     = "0774.93057",
 zblreviewer   = "T. Duncan (Lawrence)",
 doi           = "10.1016/0898-1221(93)90214-G",
 url           = "https://www.sciencedirect.com/science/article/pii/089812219390214G",
 annote        = "Complex systems tend to be desynchronized, as part and
                  parcel of their internal organization and internal
                  connections. One way that this may arise is from quite
                  small mismatching of operating times of system components.
                  On the other hand, lack of synchronization can be built
                  into the system. For example, ``chaotic'' iterations and
                  asynchronous algorithms are exploited in parallel
                  processing. Too, a system may be susceptible of change by
                  numerous factors at different, asynchronous times. In all
                  cases, it is important to understand the effect that
                  desynchronization can have on the stability of the system
                  and the convergence properties of processes.\par We
                  consider the symbolic dynamics of desynchronized switching
                  times and extract a numerical quantity whose values
                  determine the stability characteristics of the system. An
                  important role in the proof is played by Hilbert's
                  projective metric.",
}

@INPROCEEDINGS{ACDDHK:STACS15,
 author        = "Asarin, Eugene and Cervelle, Julien and Degorre, Aldric and
                  Dima, Catalin and Horn, Florian and Kozyakin, Victor",
 editor        = "Ollinger, Nicolas and Vollmer, Heribert",
 title         = "Entropy Games and Matrix Multiplication Games",
 booktitle     = "33rd Symposium on Theoretical Aspects of Computer Science,
                  (STACS 2016)",
 series        = "LIPIcs. Leibniz Int. Proc. Inform.",
 publisher     = "Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik",
 address       = "Dagstuhl, Germany",
 year          = "2016",
 volume        = "47",
 pages         = "11:1--11:14",
 isbn          = "978-3-95977-001-9",
 issn          = "1868-8969",
 mrclass       = "91A10 (15A99 68Q25)",
 mrnumber      = "3539108",
 zblnumber     = "1390.91016",
 hal_id        = "hal-01164086",
 hal_version   = "v3",
 doi           = "10.4230/LIPIcs.STACS.2016.11",
 url           = "https://drops.dagstuhl.de/opus/volltexte/2016/5712/",
 language      = "english",
 annote        = "Two intimately related new classes of games are introduced
                  and studied: entropy games (EGs) and matrix multiplication
                  games (MMGs). An EG is played on a finite arena by
                  two-and-a-half players: Despot, Tribune and the
                  non-deterministic People. Despot wants to make the set of
                  possible People's behaviors as small as possible, while
                  Tribune wants to make it as large as possible. An MMG is
                  played by two players that alternately write matrices from
                  some predefined finite sets. One wants to maximize the
                  growth rate of the product, and the other to minimize it.
                  We show that in general MMGs are undecidable in quite a
                  strong sense. On the positive side, EGs correspond to a
                  subclass of MMGs, and we prove that such MMGs and EGs are
                  determined, and that the optimal strategies are simple. The
                  complexity of solving such games is in
                  $\mathsf{NP}\cap\mathsf{coNP}$.",
}

@INCOLLECTION{AKKKP:88,
 author        = "Asarin, {\relax{}Ye}. A. and Kozyakin, V. S. and
                  Krasnosel'ski{\u\i}, M. A. and Kuznetsov, N. A. and
                  Pokrovski{\u\i}, A. V.",
 authauthor    = "Asarin, E. and Kozyakin, Victor and Krasnosel'skii, Mark and
                  Kuznetsov, N. and Pokrovskii, Alexei",
 title         = "On some new types of mathematical models of complex
                  systems",
 booktitle     = "Modelling and adaptive control ({S}opron, 1986)",
 series        = "Lecture Notes in Control and Inform. Sci.",
 publisher     = "Springer",
 address       = "Berlin",
 year          = "1988",
 volume        = "105",
 pages         = "10--26",
 mrclass       = "93A10 (93C55 93D20)",
 mrnumber      = "958694",
 zblnumber     = "0648.93025",
 zblreviewer   = "A. Van{\v{e}}{\v{c}}ek",
 doi           = "10.1007/BFb0043174",
 url           = "https://link.springer.com/chapter/10.1007/BFb0043174",
 annote        = "The paper is aimed at consideration of two new models whose
                  study has just begun. Desynchronized linear models are
                  introduced as discrete linear models with the state
                  coordinates changing at different times. Desynchronization
                  is suggested as the easiest way to attain stability for
                  some systems. Moreover, the limit hysteresis nonlinearities
                  are introduced and the averaging principle is studied.",
}

@ARTICLE{ASJ:IEEECSL17,
 author        = "Athanasopoulos, N. and Smpoukis, K. and Jungers, R. M.",
 authauthor    = "Athanasopoulos, N. and Smpoukis, K. and Jungers, R.",
 title         = "Invariant sets analysis for constrained switching systems",
 journal       = "IEEE Control Systems Letters",
 year          = "2017",
 volume        = "1",
 number        = "2",
 pages         = "256--261",
 month         = oct,
 issn          = "2475-1456",
 mrnumber      = "4208542",
 doi           = "10.1109/LCSYS.2017.2714840",
 url           = "https://ieeexplore.ieee.org/document/7946088/",
 annote        = "We study discrete time linear constrained switching systems
                  with additive disturbances, in the general setting where
                  the switching acts on the system matrices, the disturbance
                  sets, and the state constraint sets. Our primary goal is to
                  extend the existing invariant set constructions when the
                  switching signal is constrained by a given automation. We
                  achieve it by working with a relaxation of invariance,
                  namely the multi-set invariance. By exploiting recent
                  results on computing the stability metrics for these
                  systems, we establish explicit bounds on the number of
                  iterations required for each construction. Last, as an
                  application, we develop new maximal invariant set
                  constructions for the case of linear systems in far fewer
                  iterations compared to the state-of-the-art.",
}

@ARTICLE{ABY:CMH10,
 author        = "Avila, Artur and Bochi, Jairo and Yoccoz, Jean-Christophe",
 title         = "Uniformly hyperbolic finite-valued {${\rm
                  SL}(2,\mathbb{R})$}-cocycles",
 journal       = "Comment. Math. Helv.",
 fjournal      = "Commentarii Mathematici Helvetici. A Journal of the Swiss
                  Mathematical Society",
 year          = "2010",
 volume        = "85",
 number        = "4",
 pages         = "813--884",
 eprinttype    = "arXiv",
 eprint        = "0808.0133",
 issn          = "0010-2571",
 mrclass       = "37D20 (15B99 37A20 37G35)",
 mrnumber      = "2718140 (2012c:37050)",
 mrreviewer    = "M{\'a}rio Bessa",
 zblnumber     = "1201.37032",
 zblreviewer   = "Andrzej Piatkowski ({\L}{\'o}d{\'z})",
 doi           = "10.4171/CMH/212",
 url           = "https://www.ems-ph.org/journals/show_abstract.php?issn=0010-2571&vol=85&iss=4&rank=4",
 annote        = "We consider finite families of $\mathit{SL}(2,\mathbb{R})$
                  matrices whose products display uniform exponential growth.
                  These form open subsets of $(\mathit{SL}(2,\mathbb{R}))^N$,
                  and we study their components, boundary, and complement. We
                  also consider the more general situation where the allowed
                  products of matrices satisfy a Markovian rule.\par More
                  precisely, $N$ will be an integer bigger than $1$, and the
                  base $X=\Sigma\subset N^{\mathbb{Z}}$ will be a transitive
                  subshift of finite type (also called topological Markov
                  chain), equipped with the shift map
                  $\sigma:\Sigma\to\Sigma$. We will only consider cocycles
                  defined by a map $A:\Sigma\to\mathit{SL}(2,\mathbb{R})$
                  depending only on the letter in position zero. The
                  parameter space will be therefore the product
                  $(\mathit{SL}(2,\mathbb{R}))^N$. The parameters
                  $(A_1,\ldots ,A_N)$ which correspond to a uniformly
                  hyperbolic cocycle form an open set $\mathcal{H}$ which is
                  the object of our study: we would like to describe its
                  boundary, its connected components, and its complement.
                  Roughly speaking, we will see that this goal is attained
                  for the full shift on two symbols, and that new phenomena
                  appear with at least $3$ symbols which make such a complete
                  description much more difficult and complicated.",
}

@ARTICLE{AvilaRob:JMD09,
 author        = "Avila, Artur and Roblin, Thomas",
 title         = "Uniform exponential growth for some {${\rm
                  SL}(2,\mathbb{R})$} matrix products",
 journal       = "J. Mod. Dyn.",
 fjournal      = "Journal of Modern Dynamics",
 year          = "2009",
 volume        = "3",
 number        = "4",
 pages         = "549--554",
 hal_id        = "hal-00365299",
 hal_version   = "v1",
 issn          = "1930-5311",
 mrclass       = "37H15 (37A05 37C85 37D05)",
 mrnumber      = "2587085 (2011f:37099)",
 mrreviewer    = "Jairo Bochi",
 zblnumber     = "1189.37060",
 doi           = "10.3934/jmd.2009.3.549",
 url           = "https://www.aimsciences.org/article/doi/10.3934/jmd.2009.3.549",
 annote        = "Given a hyperbolic matrix $H\in SL(2,\mathbb{R})$, we prove
                  that for almost every $R\in SL(2,\mathbb{R})$, any product
                  of length $n$ of $H$ and $R$ grows exponentially fast with
                  $n$ provided the matrix $R$ occurs less than
                  $o(\frac{n}{\log n\log\log n})$ times.",
}

@INPROCEEDINGS{CACJ:DLT19,
 author        = "Azfar, Umer and Catalano, Costanza and Charlier, Ludovic and
                  Jungers, Rapha{\"e}l M.",
 authauthor    = "Catalano, Costanza and Azfar, Umer and Charlier, Ludovic and
                  Jungers, Rapha{\"e}l",
 editor        = "Hofman, Piotrek and Skrzypczak, Micha{\l}",
 title         = "A Linear Bound on the $k$-Rendezvous Time for Primitive Sets
                  of {NZ} Matrices",
 booktitle     = "Developments in language theory",
 series        = "Lecture Notes in Comput. Sci.",
 publisher     = "Springer, Cham",
 year          = "2019",
 volume        = "11647",
 pages         = "59--73",
 eprinttype    = "arXiv",
 eprint        = "1903.10421",
 isbn          = "978-3-030-24886-4",
 mrclass       = "68Q45",
 mrnumber      = "3991875",
 zblnumber     = "07117536",
 doi           = "10.1007/978-3-030-24886-4_4",
 url           = "https://link.springer.com/chapter/10.1007/978-3-030-24886-4_4",
 annote        = "A set of nonnegative matrices is called primitive if there
                  exists a product of these matrices that is entrywise
                  positive. Motivated by recent results relating
                  synchronizing automata and primitive sets, we study the
                  length of the shortest product of a primitive set having a
                  column or a row with $k$ positive entries (the $k$-RT). We
                  prove that this value is at most linear w.r.t. the matrix
                  size n for small $k$, while the problem is still open for
                  synchronizing automata. We then report numerical results
                  comparing our upper bound on the $k$-RT with heuristic
                  approximation methods.",
}

@INPROCEEDINGS{BCK:AIP16,
 author        = "Babiarz, Artur and Czornik, Adam and Klamka, Jerzy",
 title         = "A generalization of {Y}amamoto's theorem relating eigenvalue
                  moduli and singular values of a matrix",
 booktitle     = "AIP Conference Proceedings {\textbf{1738}}, 130006",
 publisher     = "AIP Publishing",
 address       = "Rhodes, Greece",
 year          = "2016",
 pages         = "130006--1--130006--4",
 doi           = "10.1063/1.4951922",
 url           = "https://aip.scitation.org/doi/abs/10.1063/1.4951922",
 annote        = "In this paper we present a generalization of Yamamoto's
                  theorem relating eigenvalue moduli and singular values of a
                  matrix. In our generalization a single matrix is replaced
                  by a bounded set of matrices. The main result of the paper
                  includes also, as a special case, the equality between
                  generalized spectral radius and joint spectral radius.",
}

@ARTICLE{BD:StochDyn21,
 author        = "Backes, Lucas and Dragi{\v{c}}evi{\'c}, Davor",
 title         = "On the spectral radius of compact operator cocycles",
 journal       = "Stoch. Dyn.",
 fjournal      = "Stochastics and Dynamics",
 year          = "2021",
 volume        = "21",
 number        = "5",
 pages         = "Paper No. 2150026, 16",
 eprinttype    = "arXiv",
 eprint        = "1810.01762",
 issn          = "0219-4937",
 mrclass       = "37H15 (15A60 37A20 37D25)",
 mrnumber      = "4297949",
 mrreviewer    = "Yongle Jiang",
 zblnumber     = "1481.37055",
 doi           = "10.1142/S021949372150026X",
 url           = "https://www.worldscientific.com/doi/abs/10.1142/S021949372150026X",
 annote        = "We extend the notions of joint and generalized spectral
                  radii to cocycles acting on Banach spaces and obtain a
                  version of Berger--Wang's formula when restricted to the
                  space of cocycles taking values in the space of compact
                  operators. Moreover, we observe that the previous
                  quantities depends continuously on the underlying
                  cocycle.",
}

@ARTICLE{BXMS:IEEETSP13,
 author        = "Bajovic, Dragana and Xavier, Joao and Moura, Jose M. F. and
                  Sinopoli, Bruno",
 authauthor    = "Bajovic, Dragana and Xavier, Joao and Moura, Jose and
                  Sinopoli, Bruno",
 title         = "Consensus and Products of Random Stochastic Matrices:
                  {E}xact Rate for Convergence in Probability",
 journal       = "IEEE Trans. Signal Process.",
 fjournal      = "IEEE Transactions on Signal Processing",
 year          = "2013",
 volume        = "61",
 number        = "10",
 pages         = "2557--2571",
 eprinttype    = "arXiv",
 eprint        = "1202.6389",
 issn          = "1053-587X",
 mrclass       = "94C15 (60B20 60F10 90B18 94A13)",
 mrnumber      = "3053826",
 mrreviewer    = "Guangyu Yang",
 zblnumber     = "1393.90025",
 doi           = "10.1109/TSP.2013.2248003",
 url           = "https://ieeexplore.ieee.org/document/6466430",
 annote        = "Distributed consensus and other linear systems with system
                  stochastic matrices $W_k$ emerge in various settings, like
                  opinion formation in social networks, rendezvous of robots,
                  and distributed inference in sensor networks. The matrices
                  $W_k$ are often random, due to, e.g., random packet
                  dropouts in wireless sensor networks. Key in analyzing the
                  performance of such systems is studying convergence of
                  matrix products $W_kW_{k-1} \cdots W_1$. In this paper, we
                  find the exact exponential rate $I$ for the convergence in
                  probability of the product of such matrices when time $k$
                  grows large, under the assumption that the $W_k$'s are
                  symmetric and independent identically distributed in time.
                  Further, for commonly used random models like with gossip
                  and link failure, we show that the rate $I$ is found by
                  solving a min-cut problem and, hence, easily computable.
                  Finally, we apply our results to optimally allocate the
                  sensors' transmission power in consensus+innovations
                  distributed detection.\par Our analysis reveals that the
                  exponential rate of convergence in probability depends only
                  on the statistics of the support graphs of the random
                  matrices. Further, we show how to compute this rate for
                  commonly used random models: gossip and link failure. With
                  these models, the rate is found by solving a min-cut
                  problem, and hence it is easily computable. Finally, as an
                  illustration, we apply our results to solving power
                  allocation among networked sensors in a
                  consensus+innovations distributed detection problem.",
}

@INCOLLECTION{BGP:ICALP17,
 author        = "Balle, Borja and Gourdeau, Pascale and Panangaden, Prakash",
 editor        = "Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian
                  and Muscholl, Anca",
 title         = "Bisimulation metrics for weighted automata",
 booktitle     = "44th International Colloquium on Automata, Languages, and
                  Programming (ICALP 2017)",
 series        = "Leibniz International Proceedings in Informatics (LIPIcs)",
 publisher     = "Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik",
 address       = "Dagstuhl, Germany",
 year          = "2017",
 volume        = "80",
 pages         = "103:1--103:14",
 eprint        = "1702.08017",
 isbn          = "978-3-95977-041-5",
 issn          = "1868-8969",
 mrclass       = "68Q85 (68Q45)",
 mrnumber      = "3685843",
 zblnumber     = "1442.68080",
 doi           = "10.4230/LIPIcs.ICALP.2017.103",
 url           = "https://drops.dagstuhl.de/opus/volltexte/2017/7395/",
 annote        = "We develop a new bisimulation (pseudo)metric for weighted
                  finite automata (WFA) that generalizes Boreale's linear
                  bisimulation relation. Our metrics are induced by seminorms
                  on the state space of WFA. Our development is based on
                  spectral properties of sets of linear operators. In
                  particular, the joint spectral radius of the transition
                  matrices of WFA plays a central role. We also study
                  continuity properties of the bisimulation pseudometric,
                  establish an undecidability result for computing the
                  metric, and give a preliminary account of applications to
                  spectral learning of weighted automata. See also
                  \cite{BGP:IC20}.",
}

@ARTICLE{BGP:IC20,
 author        = "Balle, Borja and Gourdeau, Pascale and Panangaden, Prakash",
 title         = "Bisimulation metrics and norms for real-weighted automata",
 journal       = "Inform. and Comput.",
 fjournal      = "Information and Computation",
 year          = "2022",
 volume        = "282",
 pages         = "Paper No. 104649, 19",
 eprint        = "1702.08017",
 issn          = "0890-5401",
 mrclass       = "68Q45 (68Q85)",
 mrnumber      = "4358121",
 zblnumber     = "1483.68154",
 doi           = "10.1016/j.ic.2020.104649",
 url           = "https://www.sciencedirect.com/science/article/pii/S0890540120301371",
 annote        = "We develop a new bisimulation (pseudo)metric for weighted
                  finite automata (WFA) that generalizes Boreale's linear
                  bisimulation relation. Our metrics are induced by seminorms
                  on the state space of WFA. Our development is based on
                  spectral properties of sets of linear operators. In
                  particular, the joint spectral radius of the transition
                  matrices of WFA plays a central role. We also study
                  continuity properties of the bisimulation pseudometric,
                  establish an undecidability result for computing the
                  metric, and give a preliminary account of applications to
                  spectral learning of weighted automata. See also
                  \cite{BGP:ICALP17}.",
}

@ARTICLE{Bapat:LAA98,
 author        = "Bapat, R. B.",
 authauthor    = "Bapat, R.",
 title         = "A max version of the {P}erron-{F}robenius theorem",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 booktitle     = "Proceedings of the {S}ixth {C}onference of the
                  {I}nternational {L}inear {A}lgebra {S}ociety ({C}hemnitz,
                  1996)",
 year          = "1998",
 volume        = "275/276",
 pages         = "3--18",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15A48 (15A18)",
 mrnumber      = "1628380 (99d:15017)",
 mrreviewer    = "Marc Artzrouni",
 zblnumber     = "0941.15020",
 zblreviewer   = "David Steinsaltz (Berkeley)",
 doi           = "10.1016/S0024-3795(97)10057-X",
 url           = "https://www.sciencedirect.com/science/article/pii/S002437959710057X",
 annote        = "In one variant of max algebra, the two binary operations are
                  multiplication and maximization. There is a corresponding
                  matrix theory, wherein the maximum takes the place of the
                  sum in all matrix operations. In this context one can speak
                  of maxeigenvalues and maxeigenvectors, and the purpose of
                  this (largely expository) paper is to put some
                  already-known maxeigentheory into a coherent framework, and
                  to trace out more clearly its connections to the
                  traditional eigentheory. Of particular concern is the max
                  version of the Perron--Frobenius theorem:\par
                  \textbf{Theorem.} \emph{For any nonnegative irreducible
                  matrix $A=(a_{ij})$, there is a positive vector $x$ such
                  that $\max_{j}a_{ij}x_{j}= \mu(A)x_{i}$ for
                  $i=1,2,\ldots,n$, where $\mu(A)$ is the maximum geometric
                  mean of a circuit in the weighted directed graph
                  corresponding to $A$.}\par This theorem, which we refer to
                  as the max version of the Perron--Frobenius Theorem, is
                  well-known in the context of matrices over the max algebra
                  and also in the context of matrix scalings. In the present
                  work, which is partly expository, we bring out the intimate
                  connection between this result and the Perron--Frobenius
                  theory. We present several proofs of the result, some of
                  which use the Perron--Frobenius Theorem. Structure of max
                  eigenvalues and max eigenvectors is described. Possible
                  ways to unify the Perron--Frobenius Theorem and its max
                  version are indicated. Some inequalities for $\mu(A)$ are
                  proved.\par The paper offers five different proofs of this
                  result, some of them new, reflecting the various methods of
                  proof which have been applied to the classical
                  Perron--Frobenius theorem. It also presents two different
                  generalizations which unify the traditional and the max
                  forms of this theorem. One of these, in particular, allows
                  for an interpolation between the two by defining a matrix
                  product with the traditional $\sum^{n}_{j=1}a_{ij}b_{jr}$
                  replaced by the sum of the $k$ largest products
                  $a_{ij}b_{jr}$, for $k$ a number between $1$ and $n$. If
                  $k$ is $1$, this is the max product, while $k=n$ yields the
                  traditional product.",
}

@ARTICLE{Bar:AIT88-5:e,
 author        = "Barabanov, N. E.",
 authauthor    = "Barabanov, N.",
 title         = "The {L}yapunov indicator of discrete inclusions. {III}",
 journal       = "Autom. Remote Control",
 fjournal      = "Automation and Remote Control",
 year          = "1988",
 volume        = "49",
 number        = "5",
 pages         = "558--565",
 issn          = "0005-1179",
 mrclass       = "34D05 (34A60 39A12 93C55)",
 mrnumber      = "952665 (89m:34060)",
 mrreviewer    = "J. W. Macki",
 zblnumber     = "0665.93045",
 zblreviewer   = "O. Pastravanu",
 annote        = "Certain algebraic formulas characterizing the Lyapunov
                  indicator $\rho$ of a discrete inclusion are presented. For
                  $\rho <0$ it is shown that the inclusion state space can be
                  embedded in a space of larger dimension in which the cubic
                  norm decreases along any solution of the inclusion that
                  corresponds to the original one. A similar necessary and
                  sufficient condition is derived for the case $\rho\ge 0$
                  too. An algorithm which allows to determine the sign of
                  $\rho$ in a finite number of steps is also formulated.",
}

@ARTICLE{Bar:AIT88-3:e,
 author        = "Barabanov, N. E.",
 authauthor    = "Barabanov, N.",
 title         = "The {L}yapunov indicator of discrete inclusions. {II}",
 journal       = "Autom. Remote Control",
 fjournal      = "Automation and Remote Control",
 year          = "1988",
 volume        = "49",
 number        = "3",
 pages         = "283--287",
 issn          = "0005-1179",
 mrclass       = "34A60 (34D05 39A12)",
 mrnumber      = "943889 (89e:34026)",
 mrreviewer    = "J. W. Macki",
 zblnumber     = "0665.93044",
 zblreviewer   = "O. Pastravanu",
 annote        = "The concept of conjugate inclusion is formulated and its
                  properties are studied. The problem of determination of the
                  Lyapunov indicator for an original inclusion is reduced to
                  the analysis of a conjugate autonomous system of discrete
                  equations.",
}

@ARTICLE{Bar:AIT88-2:e,
 author        = "Barabanov, N. E.",
 authauthor    = "Barabanov, N.",
 title         = "{L}yapunov indicator of discrete inclusions. {I}",
 journal       = "Autom. Remote Control",
 fjournal      = "Automation and Remote Control",
 year          = "1988",
 volume        = "49",
 number        = "2",
 pages         = "152--157",
 issn          = "0005-1179",
 mrclass       = "34A60 (34D05 39A12)",
 mrnumber      = "940263 (89e:34025)",
 mrreviewer    = "J. W. Macki",
 zblnumber     = "0665.93043",
 zblreviewer   = "O. Pastravanu",
 annote        = "An equivalence is proved between the negativity of the,
                  Lyapunov indicator corresponding to a discrete inclusion
                  defined by $F(y)=\{Ay$, $A\in \mathcal{U}\}$, a polyhedral
                  set: \[ \mathcal{U}=\{A+\sum^{m}_{i=1}b_i\nu_ ic^T_i,\quad
                  0\le \nu_i\le \mu_i,\quad b_ i,c_i\in {\mathbb{R}}^n\},\]
                  and the absolute stability of the discrete control system:
                  \[ x_{k+1}=Ax_k+\sum^{m}_{i=1}b_i\xi^i_ k,\quad
                  \sigma^i_k=c^T_ix_k,\quad \xi^ i_k=\phi_i(\sigma^i_k,k) \]
                  in a class of nonlinearities that satisfy the conditions
                  $0\le \phi_ i(\sigma,k)\sigma \le \mu_i\sigma^2$ for any
                  $\sigma\in R$, $i=1,\ldots,m$, $k=0,1,2,\ldots$. It is also
                  shown that the existence of an unbounded solution of a
                  nonsingular inclusion is equivalent to a positive Lyapunov
                  indicator.",
}

@INPROCEEDINGS{Bar:ACC95,
 author        = "Barabanov, N. E.",
 authauthor    = "Barabanov, N.",
 title         = "Stability of inclusions of linear type",
 booktitle     = "American Control Conference, Proceedings of the 1995",
 year          = "1995",
 volume        = "5",
 pages         = "3366--3370",
 month         = jun,
 doi           = "10.1109/ACC.1995.532231",
 url           = "https://ieeexplore.ieee.org/document/532231/",
 annote        = "Introduced several new concepts for the differential
                  inclusions of linear type. The main result concerns the
                  existence of an extremal norm for nonsingular inclusions.
                  It allows the author to present the numerical algorithm for
                  calculating the Lyapunov index up to arbitrary accuracy.
                  For 3-ordered automatic control systems the author
                  describes all extremal solutions and in such a way the
                  author solves the problem of absolute stability in terms of
                  coefficients of transfer functions",
}

@INPROCEEDINGS{Bar:CDC05,
 author        = "Barabanov, Nikita",
 title         = "Lyapunov Exponent and Joint Spectral Radius: Some Known and
                  New Results",
 booktitle     = "Proceedings of the 44th {IEEE} Conference on Decision and
                  Control and European Control Conference 2005, Seville,
                  Spain, December 12--15",
 year          = "2005",
 pages         = "2332--2337",
 doi           = "10.1109/CDC.2005.1582510",
 annote        = "The logarithm of joint spectral radius of a set of matrices
                  coincides with Lyapunov exponent of corresponding linear
                  inclusions. Main results about Lyapunov exponents of
                  discrete time and continuous time linear inclusions are
                  presented. They include the existence of extremal norm;
                  relations between Lyapunov indices of dual inclusions;
                  maximum principle for linear inclusions; algebraic criteria
                  for stability of linear inclusions; algorithm to find out
                  the sign of Lyapunov exponents. The main result is extended
                  to linear inclusions with delays. The Aizerman problem for
                  three-ordered timevarying continuous time systems with one
                  nonlinearity is solved. The Perron--Frobenius theorem is
                  extended for three-ordered continuous time linear
                  inclusions.",
}

@ARTICLE{BV:JGA12,
 author        = "Barnsley, Michael F. and Vince, Andrew",
 authauthor    = "Barnsley, Michael and Vince, Andrew",
 title         = "Real projective iterated function systems",
 journal       = "J. Geom. Anal.",
 fjournal      = "Journal of Geometric Analysis",
 year          = "2012",
 volume        = "22",
 number        = "4",
 pages         = "1137--1172",
 eprinttype    = "arXiv",
 eprint        = "1003.3473",
 issn          = "1050-6926",
 mrclass       = "37B25",
 mrnumber      = "2965365",
 mrreviewer    = "Jacek R. Jachymski",
 zblnumber     = "1256.28002",
 doi           = "10.1007/s12220-011-9232-x",
 url           = "https://link.springer.com/article/10.1007/s12220-011-9232-x",
 annote        = "This paper contains four main results associated with an
                  attractor of a projective iterated function system (IFS).
                  The first theorem characterizes when a projective IFS has
                  an attractor which avoids a hyperplane. The second theorem
                  establishes that a projective IFS has at most one
                  attractor. In the third theorem the classical duality
                  between points and hyperplanes in projective space leads to
                  connections between attractors that avoid hyperplanes and
                  repellers that avoid points, as well as hyperplane
                  attractors that avoid points and repellers that avoid
                  hyperplanes. Finally, an index is defined for attractors
                  which avoid a hyperplane. This index is shown to be a
                  nontrivial projective invariant.",
}

@ARTICLE{BarVin:LAA11,
 author        = "Barnsley, Michael and Vince, Andrew",
 title         = "The eigenvalue problem for linear and affine iterated
                  function systems",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2011",
 volume        = "435",
 number        = "12",
 pages         = "3124--3138",
 eprinttype    = "arXiv",
 eprint        = "1004.5040",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "28A80 (15A18 34D20 93B60)",
 mrnumber      = "2831601 (2012h:28014)",
 mrreviewer    = "Nikita E. Barabanov",
 zblnumber     = "1229.15008",
 zblreviewer   = "Qing-Wen Wang (Shanghai)",
 doi           = "10.1016/j.laa.2011.05.011",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379511004162",
 annote        = "The eigenvalue problem for a linear function $L$ centers on
                  solving the eigen-equation $Lx = \lambda \, x$. This paper
                  generalizes the eigenvalue problem from a single linear
                  function to an iterated function system $F$ consisting of
                  possibly an infinite number of linear or affine functions.
                  The eigen-equation becomes $F(X) = \lambda \, X$, where
                  $\lambda >0$ is real, $X$ is a compact set, and $F(X) =
                  \bigcup_{f\in F} f(X)$. The main result is that an
                  irreducible, linear iterated function system $F$ has a
                  unique eigenvalue $\lambda$ equal to the joint spectral
                  radius of the functions in $F$ and a corresponding eigenset
                  $S$ that is centrally symmetric, star-shaped, and full
                  dimensional. Results of Barabanov and of
                  Dranishnikov--Konyagin--Protasov on the joint spectral
                  radius follow as corollaries.",
}

@ARTICLE{Baudet:JASSCM78,
 author        = "Baudet, G{\'e}rard M.",
 authauthor    = "Baudet, G{\'e}rard",
 title         = "Asynchronous iterative methods for multiprocessors",
 journal       = "J. Assoc. Comput. Mach.",
 fjournal      = "Journal of the Association for Computing Machinery",
 year          = "1978",
 volume        = "25",
 number        = "2",
 pages         = "226--244",
 issn          = "0004-5411",
 mrclass       = "65H05",
 mrnumber      = "0494894 (58 \#13674)",
 zblnumber     = "0372.68015",
 doi           = "10.1145/322063.322067",
 url           = "https://dl.acm.org/doi/10.1145/322063.322067",
 annote        = "A class of asynchronous iterative methods is presented for
                  solving a system of equations. Existing iterative methods
                  are identified in terms of asynchronous iterations, and new
                  schemes are introduced corresponding to a parallel
                  implementation on a multiprocessor system with no
                  synchronization between cooperating processes. A sufficient
                  condition is given to guarantee the convergence of any
                  asynchronous iterations, and results are extended to
                  include iterative methods with memory. Asynchronous
                  iterative methods are then evaluated from a computational
                  point of view, and bounds are derived for the efficiency.
                  The bounds are compared with actual measurements obtained
                  by running various asynchronous iterations on a
                  multiprocessor, and the experimental results show clearly
                  the advantage of purely asynchronous iterative methods.",
}

@ARTICLE{Bell:LAA05,
 author        = "Bell, Jason P.",
 authauthor    = "Bell, Jason",
 title         = "A gap result for the norms of semigroups of matrices",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2005",
 volume        = "402",
 pages         = "101--110",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15A60 (15A18 15A42)",
 mrnumber      = "2141076 (2006c:15044)",
 mrreviewer    = "Jorma Kaarlo Merikoski",
 zblnumber     = "1074.15033",
 zblreviewer   = "Alexey Alimov (Moskva)",
 doi           = "10.1016/j.laa.2004.12.007",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379504005300",
 annote        = "Let $\|\cdot\|$ be a matrix norm on $M_d(\mathbb{C})$ and,
                  $\mathcal{A}=\{A_1,\ldots, A_e\}$ be a finite set of
                  matrices in $M_d(\mathbb{C})$. Set \[m_n(\mathcal{A})=
                  \max_{1\leq i_1,\ldots,i_n\leq e}\|A_{i_1}\cdots
                  A_{i_n}\|.\] The author shows that there is a gap in the
                  possible growth of $m_n(\mathcal{A})$. The main result
                  states:\par \emph{Let $\mathcal{A}$ be given. Then either
                  there is some constant $c>1$ such that
                  $m_n(\mathcal{A})>c^n$ for all $n$ sufficiently large, or
                  $m_n(\mathcal{A})=O(n^{d-1})$. Moreover,
                  $m_n(\mathcal{A})=O(n^{d-1})$ if and only if the
                  eigenvalues of every matrix in the semigroup generated by
                  $\mathcal{A}$ are all on or inside the unit circle}.\par As
                  a consequence of this result, it follows, for example, that
                  it is impossible to find a set of matrices $\mathcal{A}$
                  with $m_n(\mathcal{A})\sim\exp(\sqrt{n})$.",
}

@ARTICLE{BCH:SF16,
 author        = "Bell, Jason P. and Coons, Michael and Hare, Kevin G.",
 authauthor    = "Bell, Jason and Coons, Michael and Hare, Kevin",
 title         = "Growth degree classification for finitely generated
                  semigroups of integer matrices",
 journal       = "Semigroup Forum",
 fjournal      = "Semigroup Forum",
 year          = "2016",
 volume        = "92",
 number        = "1",
 pages         = "23--44",
 eprinttype    = "arXiv",
 eprint        = "1410.5519",
 issn          = "0037-1912",
 mrclass       = "20M05",
 mrnumber      = "3448399",
 zblnumber     = "1356.20034",
 zblreviewer   = "Jaak Henno",
 doi           = "10.1007/s00233-015-9725-1",
 url           = "https://link.springer.com/article/10.1007/s00233-015-9725-1",
 annote        = "Let $\mathcal{A}$ be a finite set of $d\times d$ matrices
                  with integer entries and let $m_n(\mathcal{A})$ be the
                  maximum norm of a product of $n$ elements of $\mathcal{A}$.
                  In this paper, we classify gaps in the growth of
                  $m_n(\mathcal{A})$; specifically, we prove that
                  $\lim_{n\to\infty} \log m_n(\mathcal{A})/\log
                  n\in\mathcal{B}{Z}_{\geqslant 0}\cup\{\infty\}.$ This has
                  applications to the growth of regular sequences as defined
                  by Allouche and Shallit.",
}

@ARTICLE{BerWang:LAA92,
 author        = "Berger, Marc A. and Wang, Yang",
 authauthor    = "Berger, Marc and Wang, {\relax{}Yang}",
 title         = "Bounded semigroups of matrices",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "1992",
 volume        = "166",
 pages         = "21--27",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15A30 (20M20)",
 mrnumber      = "1152485 (92m:15012)",
 mrreviewer    = "Fred W. Roush",
 zblnumber     = "0818.15006",
 doi           = "10.1016/0024-3795(92)90267-E",
 url           = "https://www.sciencedirect.com/science/article/pii/002437959290267E",
 annote        = "In this note two conjectures of \emph{I.~Daubechies} and
                  \emph{J.\,C.~Lagarias}~\cite{DaubLag:LAA92} are proved. The
                  first asserts that if $\Sigma$ is a bounded set of matrices
                  such that all left infinite products converge, then
                  $\Sigma$ generates a bounded semigroup. The second asserts
                  the equality of two differently defined joint spectral
                  radii for a bounded set of matrices. One definition
                  involves the conventional spectral radius, and one involves
                  the operator norm.",
}

@ARTICLE{LagWang:LAA95,
 author        = "Berger, Marc A. and Wang, Yang",
 authauthor    = "Berger, Marc and Wang, {\relax{}Yang}",
 title         = "The finiteness conjecture for the generalized spectral
                  radius of a set of matrices",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "1995",
 volume        = "214",
 pages         = "17--42",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15A60",
 mrnumber      = "1311628 (95k:15038)",
 mrreviewer    = "Kent Kidman",
 zblnumber     = "0818.15007",
 zblreviewer   = "E. Kreyszig (Ottawa)",
 doi           = "10.1016/0024-3795(93)00052-2",
 url           = "https://www.sciencedirect.com/science/article/pii/0024379593000522",
 annote        = "Let $\Sigma = \{A_1,A_2,\ldots\}$ be a set of $n\times n$
                  matrices, $\rho(A_j)$ the spectral radius of $A_j$,
                  $\overline\rho_k (\Sigma) = \sup \rho(A_1\cdots A_k)$,
                  $\overline\rho(\Sigma) =
                  \lim_{k\to\infty}\sup\overline\rho_k(\Sigma)^{1/k}$ the
                  ``generalized spectral radius'' (cf. \emph{I.~Doubechies}
                  and \emph{J.\,C.~Lagarias}~\cite{LagWang:LAA95})
                  $\widehat\rho_k (\Sigma) = \sup\|A_1\cdots A_k\|$,
                  $\widehat\rho(\Sigma) = \lim_{k\to\infty}\sup\widehat\rho_k
                  (\Sigma)^{1/k}$ the ``joint spectral radius'' (cf.
                  \emph{G.-C.~Rota} and
                  \emph{G.~Strang}~\cite{RotaStr:IM60}). Then $\widehat\rho =
                  \overline\rho$ for a finite set $\Sigma$ (cf.
                  \emph{M.\,A.~Berger} and
                  \emph{Y.~Wang}~\cite{BerWang:LAA92}). Finiteness conjecture
                  (FC): If $\Sigma$ is finite, there is a $k$ such that
                  $\widehat\rho(\Sigma) = \overline\rho(\Sigma) =
                  \overline\rho_k (\Sigma)^{1/k}$. Normed finiteness
                  conjecture (NFC) for a given operator norm: If $\Sigma$ is
                  finite and $\|A_j\|_{\text{op}}\le 1$, then either
                  $\widehat\rho(\Sigma)< 1$ or $\widehat\rho(\Sigma) =
                  \overline\rho(\Sigma) = \overline\rho_k (\Sigma)^{1/k} = 1$
                  for some $k$.\par Results in this paper: FC is true iff NFC
                  is true for all operator norms. NFC is proved for a
                  relatively large class of operator norms. For polytope
                  norms and for the Euclidean norm, explicit upper bounds are
                  given for the least $k$ having $\overline\rho(\Sigma) =
                  \overline\rho_k (\Sigma)^{1/k}$, implying upper bounds for
                  generalized critical exponents for these norms.",
}

@BOOK{BertTsi:89,
 author        = "Bertsekas, D. P. and Tsitsiklis, J. N.",
 authauthor    = "Bertsekas, D. and Tsitsiklis, J.",
 title         = "Parallel and Distributed Computation. {N}umerical Methods",
 publisher     = "Prentice Hall",
 address       = "Englewood Cliffs. NJ",
 year          = "1989",
 pagetotal     = "715",
 isbn          = "978-1-886529-15-1",
 mrclass       = "65Y05 (65Fxx 65Hxx 68M14 68W10 90C39)",
 mrnumber      = "3587745",
 zblnumber     = "0743.65107",
 zblreviewer   = "M. Vajter{\'s}ic (Bratislava)",
 annote        = "Parallel processing has confirmed its usefulness already by
                  solving a number of important problems of numerical
                  mathematics. From advent of parallel computers these have
                  been used for a broad variety of large-sized and time
                  demanding applications. Model numerical problems serve in
                  benchmarks to test every new parallel computer system
                  appearing on the market.\par From a mathematical point of
                  view, the monograph deals with analysis, design and
                  implementation of parallel numerical algorithms. On a space
                  of more than one hundred pages an adequate introductory
                  knowledge about general aspects of parallel processing is
                  given. The kernel of the book is structured into two main
                  parts. The first part is devoted to synchronous algorithms
                  while the second one is oriented towards methods which are
                  based on the asynchronous execution principle.\par There
                  are five chapters which describe parallel synchronous
                  algorithms. A detailed attention is paid to solving linear
                  systems of equations with general and special matrices in
                  the first chapter of them. The next chapter treats
                  nonlinear problems. The synchronous part concludes chapters
                  for the shortest path problem, dynamic programming and
                  network flow analysis.\par A class of totally asynchronous
                  methods with some theoretical formulations is the subject
                  of the first chapter in the asynchronous part of the book.
                  The subsequent chapter is devoted to partially asynchronous
                  algorithms, i.e. when some amount of synchronization is
                  introduced in the algorithm. The final chapter of the
                  monograph presents a design of an asynchronous network of
                  processors for realization of a general type of parallel
                  algorithms.\par It is to recommend the book to those
                  readers who deal seriously with solving numerical problems
                  on advanced computers with parallel architecture. It is not
                  just a ``cooking book'' containing algorithmic recipes but
                  there can be found enough from the theory of parallel
                  numerics itself.",
}

@ARTICLE{BE:EJLA97,
 author        = "Beyn, W.-J. and Elsner, L.",
 title         = "Infinite products and paracontracting matrices",
 journal       = "Electron. J. Linear Algebra",
 fjournal      = "Electronic Journal of Linear Algebra",
 year          = "1997",
 volume        = "2",
 pages         = "1--8 (electronic)",
 issn          = "1081-3810",
 mrclass       = "65F10",
 mrnumber      = "1443067 (98a:65036)",
 mrreviewer    = "L. W. Ehrlich",
 zblnumber     = "0893.65019",
 zblreviewer   = "J. P. Milaszewicz (Dublin)",
 doi           = "10.13001/1081-3810.1006",
 url           = "https://journals.uwyo.edu/index.php/ela/article/view/13",
 annote        = "Infinite products of matrices arise naturally when chaotic
                  iteration procedures are applied to solve consistent linear
                  systems. The convergence of such products, when each factor
                  belongs to a finite set of matrices $\Sigma$, is ensured
                  when they are paracontracting with respect to a fixed norm
                  in the underlying vector space. The paper addresses the
                  reciprocal statement and a general affirmative answer is
                  given if a supplementary hypothesis is satisfied, namely
                  that the set of fixed points for each matrix in $\Sigma$ is
                  the same for all of them. This hypothesis is shown to be
                  equivalent to the continuity of the function that
                  associates to each sequence of indices varying in the index
                  set of $\Sigma$, the corresponding matrix product limit.
                  For the particular case where $\Sigma$ consists of two
                  projectors, the equivalence is established without the
                  supplementary hypothesis.",
}

@TECHREPORT{BeynEls96,
 author        = "Beyn, Wolf-J{\"u}rgen and Elsner, Ludwig",
 title         = "Connecting paracontractivity and convergence of products",
 institution   = "Universit{\"a}t Bielefeld",
 year          = "1996",
 number        = "96-074",
 type          = "Preprint // Sonderforschungsbereich 343, Diskrete Strukturen
                  in der Mathematik",
 url           = "https://www.math.uni-bielefeld.de/~beyn/AG_Numerik/html/en/preprints/sfb_01_09.html",
 annote        = "In~\cite{DaubLag:LAA92} the LCP-property of a finite set
                  $\Sigma$ of square complex matrices was introduced and
                  studied by \emph{I.~Daubechies} and \emph{J.\,C.~Lagarias}.
                  $\Sigma$ is an LCP-set if all left infinite products formed
                  from matrices in $\Sigma$ are convergent. It had been shown
                  earlier that a set $\Sigma$ paracontracting with respect to
                  a fixed norm is an LCP-set. Here we prove a converse
                  statement If $\Sigma$ is an LCP-set with a continuous limit
                  function then there exists a norm such that all matrices in
                  $\Sigma$ are paracontracting with respect to this norm. In
                  addition we introduce the stronger property of
                  l-paracontractivity. It is shown that common
                  l-paracontractivity of a set of matrices has a simple
                  characterization. It turns out that in the above mentioned
                  converse statement the norm can be chosen such that all
                  matrices are l-paracontracting. It is shown that for
                  $\Sigma$ consisting of two projectors the LCP-property is
                  equivalent to l-paracontractivity, even without requiring
                  continuity.",
}

@ARTICLE{BhaBha:SM95,
 author        = "Bhatia, Rajendra and Bhattacharyya, Tirthankar",
 title         = "On the joint spectral radius of commuting matrices",
 journal       = "Studia Math.",
 fjournal      = "Studia Mathematica",
 year          = "1995",
 volume        = "114",
 number        = "1",
 pages         = "29--38",
 issn          = "0039-3223",
 mrclass       = "47A13 (15A18)",
 mrnumber      = "1330215 (96f:47011)",
 mrreviewer    = "Makoto Takaguchi",
 zblnumber     = "0830.47002",
 url           = "http://matwbn.icm.edu.pl/ksiazki/sm/sm114/sm11413.pdf",
 annote        = "For a commuting $n$-tuple of matrices we introduce the
                  notion of a joint spectral radius with respect to the
                  $p$-norm and prove a spectral radius formula.",
}

@INCOLLECTION{BJ:CANT10,
 author        = "Blondel, V. D. and Jungers, R. M.",
 authauthor    = "Vincent Blondel and Raphael Jungers",
 title         = "Long products of matrices",
 booktitle     = "Combinatorics, {A}utomata and {N}umber {T}heory",
 series        = "Encyclopedia of Mathematics and its Applications",
 chapter       = "11",
 publisher     = "Cambridge University Press",
 address       = "Cambridge",
 year          = "2010",
 number        = "135",
 pages         = "530--562",
 isbn          = "9780-521-51597-9",
 doi           = "10.1017/CBO9780511777653.012",
 url           = "https://dx.doi.org/10.1017/CBO9780511777653.012",
 annote        = "The \emph{joint spectral radius} of a set of matrices is the
                  maximal growth rate that can be obtained by forming long
                  products of matrices taken in the set. This quantity and
                  its minimal growth counterpart, the \emph{joint spectral
                  subradius}, have proved useful for studying several
                  problems from combinatorics and number theory. For
                  instance, they characterize the growth of certain classes
                  of languages, the capacity of forbidden difference
                  constraints on languages, and the trackability of sensor
                  networks. In Section 11.2 we describe some of these
                  applications.\par While the joint spectral radius and
                  related notions have applications in combinatorics and
                  number theory, these disciplines have in turn been helpful
                  to improve our understanding of problems related to the
                  joint spectral radius. As an example, we present in Section
                  11.3 a central result that has been proved with the help of
                  techniques from combinatorics on words: the falseness of
                  the finiteness conjecture.\par In practice, computing a
                  joint spectral radius is not an easy task. As we will see,
                  this quantity is NP-hard to approximate in general, and the
                  simple question of knowing, given a set of matrices, if its
                  joint spectral radius is larger than one is even
                  algorithmically undecidable. However, in recent years,
                  approximation algorithms have been proposed that perform
                  well in practice. Some of these algorithms run in
                  exponential time while others provide no accuracy
                  guarantee. In practice, by combining the advantages of the
                  different algorithms, it is often possible to obtain
                  satisfactory estimates.",
}

@INPROCEEDINGS{BTV:MTNS02,
 author        = "Blondel, V. D. and Theys, J. and Vladimirov, A. A.",
 authauthor    = "Blondel, V. and Theys, J. and Vladimirov, A.",
 title         = "Switched Systems that are Periodically Stable May be
                  Unstable",
 booktitle     = "Proc. of the Symposium MTNS",
 address       = "Notre-Dame, {USA}",
 year          = "2002",
 url           = "https://www3.nd.edu/~mtns/papers/10181.pdf",
 annote        = "We prove the existence of two $2\times 2$ real matrices such
                  that all periodic products of these matrices converge to
                  zero but there exists an infinite product that does not. We
                  outline implications of this result for the stability of
                  switched linear systems, and for the finiteness
                  conjecture",
}

@ARTICLE{Blondel:LAA08,
 author        = "Blondel, Vincent D.",
 authauthor    = "Blondel, Vincent",
 title         = "The birth of the joint spectral radius: an interview with
                  {G}ilbert {S}trang",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2008",
 volume        = "428",
 number        = "10",
 pages         = "2261--2264",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "01A70 (15A18)",
 mrnumber      = "2405243",
 zblnumber     = "1149.15001",
 zblreviewer   = "Grozio Stanilov (Sofia)",
 doi           = "10.1016/j.laa.2007.12.010",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379507005708",
 annote        = "The definition of the joint spectral radius first appeared
                  in 1960. It has resurfaced in a number of application
                  areas, including wavelets, control theory, number theory,
                  coding and sensor networks. In the interview the author
                  discusses how this notion was actually born.",
}

@ARTICLE{BBKPT:TCS01,
 author        = "Blondel, Vincent D. and Bournez, Olivier and Koiran, Pascal
                  and Papadimitriou, Christos H. and Tsitsiklis, John N.",
 authauthor    = "Blondel, Vincent and Bournez, Olivier and Koiran, Pascal and
                  Papadimitriou, Christos and Tsitsiklis, John",
 title         = "Deciding stability and mortality of piecewise affine
                  dynamical systems",
 journal       = "Theoret. Comput. Sci.",
 fjournal      = "Theoretical Computer Science",
 year          = "2001",
 volume        = "255",
 number        = "1-2",
 pages         = "687--696",
 coden         = "TCSDI",
 issn          = "0304-3975",
 mrclass       = "03B25 (03D35 37B99)",
 mrnumber      = "1819103 (2002a:03021)",
 zblnumber     = "0973.68067",
 doi           = "10.1016/S0304-3975(00)00399-6",
 url           = "https://www.sciencedirect.com/science/article/pii/S0304397500003996",
 annote        = "In this paper we study problems such as: given a discrete
                  time dynamical system of the form $x(t+1)=f(x(t))$ where
                  $f:R^n \to R^n$ is a piecewise affine function, decide
                  whether all trajectories converge to 0. We show in our main
                  theorem that this attractivity problem is undecidable as
                  soon as $n\ge 2$. The same is true of two related problems:
                  Stability (is the dynamical system globally asymptotically
                  stable?) and mortality (do all trajectories go through 0?).
                  We then show that attractivity and stability become
                  decidable in dimension 1 for continuous functions.",
}

@ARTICLE{BC:TCS03,
 author        = "Blondel, Vincent D. and Canterini, Vincent",
 authauthor    = "Blondel, Vincent and Canterini, Vincent",
 title         = "Undecidable problems for probabilistic automata of fixed
                  dimension",
 journal       = "Theory Comput. Syst.",
 fjournal      = "Theory of Computing Systems",
 publisher     = "Springer US, New York, NY",
 year          = "2003",
 volume        = "36",
 number        = "3",
 pages         = "231--245",
 coden         = "TCSYFI",
 issn          = "1432-4350; 1433-0490/e",
 mrclass       = "68Q45 (03D35 60C05)",
 mrnumber      = "1962327 (2004a:68062)",
 mrreviewer    = "George Tourlakis",
 zblnumber     = "1039.68061",
 doi           = "10.1007/s00224-003-1061-2",
 url           = "https://link.springer.com/article/10.1007/s00224-003-1061-2",
 annote        = "We prove that several problems associated to probabilistic
                  finite automata are undecidable for automata whose number
                  of input letters and number of states are fixed. As a
                  corollary of one of our results we prove that the problem
                  of determining if the set of all products of two $47\times
                  47$ matrices with nonnegative rational entries is bounded
                  is undecidable.",
}

@ARTICLE{BGT:IEEETAC00,
 author        = "Blondel, Vincent D. and Gaubert, St{\'e}phane and
                  Tsitsiklis, John N.",
 authauthor    = "Blondel, Vincent and Gaubert, St{\'e}phane and Tsitsiklis,
                  John",
 title         = "Approximating the spectral radius of sets of matrices in the
                  max-algebra is {NP}-hard",
 journal       = "IEEE Trans. Automat. Control",
 fjournal      = "Institute of Electrical and Electronics Engineers.
                  Transactions on Automatic Control",
 year          = "2000",
 volume        = "45",
 number        = "9",
 pages         = "1762--1765",
 coden         = "IETAA9",
 issn          = "0018-9286",
 mrclass       = "93C65 (65Y20 68Q17)",
 mrnumber      = "1792275 (2001h:93053)",
 zblnumber     = "0990.93073",
 doi           = "10.1109/9.880644",
 url           = "https://ieeexplore.ieee.org/document/880644",
 annote        = "The lower and average spectral radii measure, respectively,
                  the minimal and average growth rates of long products of
                  matrices taken from a finite set. The logarithm of the
                  average spectral radius is traditionally called Lyapunov
                  exponent. When one performs these products in the
                  max-algebra, we obtain quantities that measure the
                  performance of discrete event systems. We show that
                  approximating the lower and average max-algebraic spectral
                  radii is NP-hard.",
}

@ARTICLE{BJO:AJIFAC15,
 author        = "Blondel, Vincent D. and Jungers, Rapha{\"e}l M. and
                  Olshevsky, Alex",
 authauthor    = "Vincent Blondel and Raphael Jungers and Alex Olshevsky",
 title         = "On primitivity of sets of matrices",
 journal       = "Automatica J. IFAC",
 fjournal      = "Automatica. A Journal of IFAC, the International Federation
                  of Automatic Control",
 year          = "2015",
 volume        = "61",
 pages         = "80--88",
 eprinttype    = "arXiv",
 eprint        = "1306.0729",
 issn          = "0005-1098",
 mrclass       = "Preliminary Data",
 mrnumber      = "3401692",
 zblnumber     = "06523630",
 doi           = "10.1016/j.automatica.2015.07.026",
 url           = "https://www.sciencedirect.com/science/article/pii/S0005109815003118",
 annote        = "A nonnegative matrix $A$ is called primitive if $A^k$ is
                  positive for some integer $k>0$. A generalization of this
                  concept to sets of matrices is the following: a set of
                  matrices $\mathcal{M} = \{A_1, A_2,\ldots, A_m \}$ is
                  primitive if $A_{i_1} A_{i_2} \cdots A_{i_k}$ is positive
                  for some indices $i_1, i_2,\ldots, i_k$. The concept of
                  primitive sets of matrices is of importance in several
                  applications, including the problem of computing the
                  Lyapunov exponents of switching systems. In this paper, we
                  analyze the computational complexity of deciding if a given
                  set of matrices is primitive and we derive bounds on the
                  length of the shortest positive product.\par We show that
                  while primitivity is algorithmically decidable, unless P=NP
                  it is not possible to decide positivity of a matrix set in
                  polynomial time. Moreover, we show that the length of the
                  shortest positive sequence can be exponential in the
                  dimension of the matrices. On the other hand, we give a
                  simple combinatorial proof of the fact that when the
                  matrices have no zero rows or zero columns, primitivity can
                  be decided in polynomial time. This latter observation is
                  related to the well-known 1964 conjecture of
                  {\v{C}}ern{\'y} on synchronizing automata. Moreover, we
                  show that for such matrices the length of the shortest
                  positive sequence is at most polynomial in the dimension.",
}

@ARTICLE{BJP:IEEETIT06,
 author        = "Blondel, Vincent D. and Jungers, Rapha{\"e}l and Protasov,
                  Vladimir",
 authauthor    = "Blondel, Vincent and Jungers, Rapha{\"e}l and Protasov,
                  Vladimir",
 title         = "On the complexity of computing the capacity of codes that
                  avoid forbidden difference patterns",
 journal       = "IEEE Trans. Inform. Theory",
 fjournal      = "Institute of Electrical and Electronics Engineers.
                  Transactions on Information Theory",
 year          = "2006",
 volume        = "52",
 number        = "11",
 pages         = "5122--5127",
 eprinttype    = "arXiv",
 eprint        = "cs/0601036",
 coden         = "IETTAW",
 issn          = "0018-9448",
 mrclass       = "94A24",
 mrnumber      = "2300380 (2007m:94086)",
 zblnumber     = "1320.94039",
 doi           = "10.1109/TIT.2006.883615",
 url           = "https://ieeexplore.ieee.org/document/1715550",
 annote        = "Some questions related to the computation of the capacity of
                  codes that avoid forbidden difference patterns are
                  analysed. The maximal number of $n$-bit sequences whose
                  pairwise differences do not contain some given forbidden
                  difference patterns is known to increase exponentially with
                  $n$; the coefficient of the exponent is the capacity of the
                  forbidden patterns. In this paper, new inequalities for the
                  capacity are given that allow for the approximation of the
                  capacity with arbitrary high accuracy. The computational
                  cost of the algorithm derived from these inequalities is
                  fixed once the desired accuracy is given. Subsequently, a
                  polynomial time algorithm is given for determining if the
                  capacity of a set is positive while the same problem is
                  shown to be NP-hard when the sets of forbidden patterns are
                  defined over an extended set of symbols. Finally, the
                  existence of extremal norms is proved for any set of
                  matrices arising in the capacity computation. Based on this
                  result, a second capacity approximating algorithm is
                  proposed. The usefulness of this algorithm is illustrated
                  by computing exactly the capacity of particular codes that
                  were only known approximately",
}

@ARTICLE{BKPW:LAA08,
 author        = "Blondel, Vincent D. and Karow, Michael and Protassov,
                  Vladimir and Wirth, Fabian R.",
 authauthor    = "Blondel, Vincent and Karow, Michael and Protasov, Vladimir
                  and Wirth, Fabian",
 title         = "Special issue on the joint spectral radius: theory, methods
                  and applications",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2008",
 volume        = "428",
 number        = "10",
 pages         = "2259--2260",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15-06",
 mrnumber      = "2405242",
 doi           = "10.1016/j.laa.2008.01.016",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379508000402",
 annote        = "The joint spectral radius was first defined in 1960 in a
                  paper by G.-C. Rota and G. Strang. It describes the maximal
                  growth rate of arbitrary matrix products of matrices from a
                  given set of matrices. As such it is a natural notion in
                  matrix analysis. It turns out that what is easy to
                  understand for a single matrix, in general poses a number
                  of subtle problems for a set of matrices even if this set
                  only contains two elements.\par In the beginning the notion
                  did not receive widespread attention, even though it is
                  intimately related to the stability of time-varying systems
                  and at the time there was considerable interest in this
                  subject. The first edition of Cesari's monograph on
                  stability of ordinary differential equations had just
                  appeared in which this problem has a prominent place. Also
                  Yoshizawa's monograph on the theory of Lyapunov functions
                  only appeared 6 years later. Indeed the intimate link of
                  the theory of the joint spectral radius with stability
                  theory and in particular Lyapunov functions was not noted
                  until in the late 80s Barabanov and also Pyatnitskii took
                  up the study of this problem independently. However, at the
                  time of the first definition of the joint spectral radius
                  this connection was unnoticed.\par In the early 90s this
                  all changed. On one hand it was observed, that the question
                  of convergence of products of matrices arises in a natural
                  way in the theory of wavelets. On the other hand
                  timevarying systems were receiving greater attention in the
                  field of control and the study of linear inclusions was
                  prominent. Driven by interest in these two fields,
                  important progress has been made during the last decade.
                  Further interesting applications of the joint spectral
                  radius have been discovered. This quantity turned out to be
                  useful in the analysis of the capacity of certain codes, in
                  the stability analysis of numerical integration schemes and
                  in the study of nonhomogeneous Markov chains. From a
                  theoretical point of view the link between the joint
                  spectral radius and extremal norms was explored in detail,
                  regularity properties of the joint spectral radius as a map
                  were discovered and approximation schemes were shown.
                  Probably the most famous results in this area are related
                  to the finiteness property, which was conjectured to hold
                  for all finite matrix sets. This conjecture was later
                  disproved by showing that a counterexample must exist.1
                  More on this subject can be found in this issue. Finally,
                  many different approaches to the calculation or maybe
                  better approximation of the joint spectral radius were
                  developed. These range from ideas using basic inequalities
                  for the joint spectral radius, the numerical construction
                  of extremal norms, the relation of the joint spectral
                  radius to higher Kronecker powers, to discounted optimal
                  control techniques based on the theory of Lyapunov
                  exponents.\par In this special issue we were fortunate to
                  obtain submissions which present advances as well in new
                  developments in the computation of the joint spectral
                  radius as in theoretical issues.\par We are grateful to all
                  the contributors to this special issue and we hope that
                  this collection will prove to be useful for researchers
                  interested in the field. Clearly, there remain many open
                  questions and new ones will arise as new applications are
                  discovered.",
}

@MISC{BN03,
 author        = "Blondel, Vincent D. and Nesterov, {\relax{}Yu}rii",
 authauthor    = "Blondel, Vincent and Nesterov, {\relax{}Yu}rii",
 title         = "Fast and precise approximations of the joint spectral
                  radius",
 howpublished  = "EUEN/CORE -- Center for operations research and
                  econometrics, UCL",
 year          = "2003",
 doi           = "10.2139/ssrn.981383",
 url           = "https://papers.ssrn.com/sol3/papers.cfm?abstract_id=981383",
 note          = "Working paper",
 annote        = "In this paper, we introduce a procedure for approximating
                  the joint spectral radius of a finite set of matrices with
                  arbitrary precision. Our approximation procedure is based
                  on semidefinite liftings and can be implemented in a
                  recursive way. For two matrices even the first step of the
                  procedure gives an approximation, whose relative quality is
                  at least $1/\sqrt{2}$, that is, more than 70\%. The
                  subsequent steps improve the quality but also increase the
                  dimension of the auxiliary problem from which this
                  approximation can be found. In an improved version of our
                  approximation procedure we show how a relative quality of
                  $(1/\sqrt{2})^{1/k}$ can be obtained by evaluating the
                  spectral radius of a single matrix of dimension $n^k (n^k
                  +1)/2$ where $n$ is the dimension of the initial matrices.
                  This result is computationally optimal in the sense that it
                  provides an approximation of relative quality
                  $1-\varepsilon$ in time polynomial in $n^{1/\varepsilon}$
                  and it is known that, unless P=NP, no such algorithm is
                  possible that runs in time polynomial in $n$ and
                  $1/\varepsilon$.\par For the special case of matrices with
                  non-negative entries we prove that
                  \[(1/2)^{1/k}\rho^{1/k}(A_1^{\otimes k}+A_2^{\otimes
                  k})\leq \rho (A_1,A_2)\leq \rho^{1/k}(A_1^{\otimes
                  k}+A_2^{\otimes k})\] where $A^{\otimes k}$ denotes the
                  $k$th Kronecker power of $A$. An approximation of relative
                  quality $(1/2)^{1/k}$ can therefore be obtained by
                  computing the spectral radius of a single matrix of
                  dimension $n^k$. From these inequalities it also follows
                  that the spectral radius is given by the simple expression
                  \[\rho (A_1, A_2)=\lim_{k\to\infty}\|A_1^{\otimes k} +
                  A_2^{\otimes k}\|^{1/k}\] where it is somewhat surprising
                  to notice that the right hand side does not directly
                  involve any mixed products between the matrices $A_1$ and
                  $A_2$.",
}

@ARTICLE{BN:SIAMJMA05,
 author        = "Blondel, Vincent D. and Nesterov, {\relax{}Yu}rii",
 authauthor    = "Blondel, Vincent and Nesterov, {\relax{}Yu}rii",
 title         = "Computationally efficient approximations of the joint
                  spectral radius",
 journal       = "SIAM J. Matrix Anal. Appl.",
 fjournal      = "SIAM Journal on Matrix Analysis and Applications",
 year          = "2005",
 volume        = "27",
 number        = "1",
 pages         = "256--272 (electronic)",
 eprint        = "math/0407485",
 issn          = "0895-4798",
 mrclass       = "15A18 (15A60 90C22)",
 mrnumber      = "2176820 (2006k:15027)",
 mrreviewer    = "Jor-Ting Chan",
 zblnumber     = "1089.65031",
 zblreviewer   = "Constantin Popa (Constanta)",
 doi           = "10.1137/040607009",
 url           = "https://epubs.siam.org/doi/10.1137/040607009",
 annote        = "The joint spectral radius of a set of matrices is a measure
                  of the maximal asymptotic growth rate that can be obtained
                  by forming long products of matrices taken from the set.
                  This quantity appears in a number of application contexts
                  but is notoriously difficult to compute and to approximate.
                  We introduce in this paper a procedure for approximating
                  the joint spectral radius of a finite set of matrices with
                  arbitrary high accuracy. Our approximation procedure is
                  polynomial in the size of the matrices once the number of
                  matrices and the desired accuracy are fixed.\par For the
                  special case of matrices with non-negative entries we give
                  elementary proofs of simple inequalities that we then use
                  to obtain approximations of arbitrary high accuracy. From
                  these inequalities it follows that the spectral radius of
                  matrices with non-negative entries is given by the simple
                  expression \[\rho(A_1,\ldots, A_m)= \lim_{k \rightarrow
                  \infty} \rho^{1/k} ( A_1^{\otimes k} + \cdots +
                  A_m^{\otimes k})\] where it is somewhat surprising to
                  notice that the right hand side does not directly involve
                  any mixed product between the matrices ($A^{\otimes k}$
                  denotes the $k$-th Kronecker power of $A$).\par For
                  matrices with arbitrary entries (not necessarily
                  non-negative) we introduce an approximation procedure based
                  on semi-definite liftings that can be implemented in a
                  recursive way. For two matrices, even the first step of the
                  procedure gives an approximation whose relative accuracy is
                  at least ${1 / \sqrt{2}}$, that is, more than $70\%$. The
                  subsequent steps improve the accuracy but also increase the
                  dimension of the auxiliary problems from which the
                  approximation can be found.\par Our approximation
                  procedures provide approximations of relative accuracy
                  $1-\epsilon$ in time polynomial in $n^{(\ln m) /\epsilon}$,
                  where $m$ is the number of matrices and $n$ is their size.
                  These bounds are close from optimality since we show that,
                  unless P=NP, no approximation algorithm is possible that
                  provides a relative accuracy of $1-\epsilon$ and runs in
                  time polynomial in $n$ and $1/\epsilon$.\par As a
                  by-product of our results we prove that a widely used
                  approximation of the joint spectral radius based on common
                  quadratic Lyapunov functions (or on ellipsoid norms) has
                  relative accuracy $1/\sqrt m$, where $m$ is the number of
                  matrices.",
}

@ARTICLE{BN:SIAMJMAA09,
 author        = "Blondel, Vincent D. and Nesterov, {\relax{}Yu}rii",
 authauthor    = "Blondel, Vincent and Nesterov, {\relax{}Yu}rii",
 title         = "Polynomial-time computation of the joint spectral radius for
                  some sets of nonnegative matrices",
 journal       = "SIAM J. Matrix Anal. Appl.",
 fjournal      = "SIAM Journal on Matrix Analysis and Applications",
 year          = "2009",
 volume        = "31",
 number        = "3",
 pages         = "865--876",
 issn          = "0895-4798",
 mrclass       = "90C22 (15A18 15A60 91B54 93C30 93C55)",
 mrnumber      = "2538656 (2010f:90111)",
 mrreviewer    = "Yves Lucet",
 zblnumber     = "1201.65051",
 zblreviewer   = "R. Militaru (Craiova)",
 doi           = "10.1137/080723764",
 url           = "https://epubs.siam.org/doi/10.1137/080723764",
 annote        = "We propose two simple upper bounds for the joint spectral
                  radius of sets of nonnegative matrices. These bounds, the
                  joint column radius and the joint row radius, can be
                  computed in polynomial time as solutions of convex
                  optimization problems. We show that these bounds are within
                  a factor $1/n$ of the exact value, where n is the size of
                  the matrices. Moreover, for sets of matrices with
                  independent column uncertainties or with independent row
                  uncertainties, the corresponding bounds coincide with the
                  joint spectral radius. In these cases, the joint spectral
                  radius is also given by the largest spectral radius of the
                  matrices in the set. As a by-product of these results, we
                  propose a polynomial-time technique for solving Boolean
                  optimization problems related to the spectral radius. We
                  also describe economics and engineering applications of our
                  results.",
}

@INCOLLECTION{BNT:LNSC04,
 author        = "Blondel, Vincent D. and Nesterov, {\relax{}Yu}rii and Theys,
                  Jacques",
 authauthor    = "Blondel, Vincent and Nesterov, {\relax{}Yu}rii and Theys,
                  Jacques",
 title         = "Approximations of the Rate of Growth of Switched Linear
                  Systems",
 booktitle     = "Hybrid Systems: {C}omputation and Control (Proceedings of
                  the 7th International Workshop, HSCC 2004, Philadelphia,
                  PA, USA, March 25-27, 2004)",
 series        = "Lecture Notes in Computer Science",
 publisher     = "Springer-Verlag",
 address       = "Berlin Heidelberg",
 year          = "2004",
 volume        = "2993",
 pages         = "173--186",
 zblnumber     = "1135.93333",
 doi           = "10.1007/978-3-540-24743-2_12",
 url           = "https://link.springer.com/chapter/10.1007/978-3-540-24743-2_12",
 annote        = "The joint spectral radius of a set of matrices is a measure
                  of the maximal asymptotic growth rate that can be obtained
                  by forming long products of matrices taken from the set.
                  This quantity appears in a number of application contexts,
                  in particular it characterizes the growth rate of switched
                  linear systems. The joint spectral radius is notoriously
                  difficult to compute and to approximate. We introduce in
                  this paper the first polynomial time approximations of
                  guaranteed precision. We provide an approximation
                  $\hat{\rho}$ that is based on ellipsoid norms that can be
                  computed by convex optimization and that is such that the
                  joint spectral radius belongs to the interval
                  $[\hat{\rho}/\sqrt{n}, \hat{\rho}]$ where n is the
                  dimension of the matrices. We also provide a simple
                  approximation for the special case where the entries of all
                  the matrices are non-negative; in this case the
                  approximation is proved to be within a factor at most $m$
                  ($m$ is the number of matrices) of the exact value.",
}

@ARTICLE{BNT:LAA05,
 author        = "Blondel, Vincent D. and Nesterov, {\relax{}Yu}rii and Theys,
                  Jacques",
 authauthor    = "Blondel, Vincent and Nesterov, {\relax{}Yu}rii and Jacques
                  Theys",
 title         = "On the accuracy of the ellipsoid norm approximation of the
                  joint spectral radius",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2005",
 volume        = "394",
 pages         = "91--107",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15A60",
 mrnumber      = "2100578 (2005i:15043)",
 mrreviewer    = "Nicola Guglielmi",
 zblnumber     = "1086.15020",
 zblreviewer   = "Shuhuang Xiang (Changsha)",
 doi           = "10.1016/j.laa.2004.06.024",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379504003003",
 annote        = "The authors consider two computable approximations of the
                  joint spectral radius for a finite set $\mathcal{U}$ of
                  matrices. The first approximation satisfies
                  $\frac{1}{\sqrt{n}}\hat{\rho}\le\rho\le\hat{\rho}$ based on
                  ellipsoid norms, where $\rho$ is the joint spectral radius
                  of $\mathcal{U}$, $\hat{\rho}=\inf_{P\succ
                  0}\max_{A_i\in\mathcal{U}}\|A_i\|_p$, $\|A\|_p$ is induced
                  by the vector norm $\|x\|_p=\sqrt{x^{T}Px}$ and $n$ is the
                  dimension of the matrices. Moreover, for the special case
                  of symmetric matrices, triangular matrices, or for sets of
                  matrices that have a solvable Lie algebra, the equality
                  $\rho=\hat{\rho}$ is satisfied. The other approximation is
                  for the set of nonnegative matrices. In this case the
                  approximation is proved to be within a factor at most $m$
                  of the exact value, where $m$ is the number of matrices in
                  $\mathcal{U}$.",
}

@ARTICLE{BTV:SIAMJMA03,
 author        = "Blondel, Vincent D. and Theys, Jacques and Vladimirov,
                  Alexander A.",
 authauthor    = "Blondel, Vincent and Theys, Jacques and Vladimirov,
                  Alexander",
 title         = "An elementary counterexample to the finiteness conjecture",
 journal       = "SIAM J. Matrix Anal. Appl.",
 fjournal      = "SIAM Journal on Matrix Analysis and Applications",
 year          = "2003",
 volume        = "24",
 number        = "4",
 pages         = "963--970 (electronic)",
 issn          = "0895-4798",
 mrclass       = "15A18 (68Q45 93B25 93D09)",
 mrnumber      = "2003315 (2004g:15010)",
 zblnumber     = "1043.15007",
 zblreviewer   = "Rodica Covaci (Cluj-Napoca)",
 doi           = "10.1137/S0895479801397846",
 url           = "https://epubs.siam.org/doi/10.1137/S0895479801397846",
 annote        = "The finiteness conjecture has recently been proved to be
                  false (cf. \emph{T.~Bousch} and
                  \emph{J.~Mairesse}~\cite{BM:JAMS02}). The present paper
                  provides an alternative proof of this fact. The authors
                  prove that there exist (infinitely many) values of the real
                  parameters $a$ and $b$ for which the matrices
                  \[a\left(\begin{array}{ll}1&1\\0&1\end{array}\right)\qquad
                  \mbox{and}\qquad
                  b\left(\begin{array}{ll}1&0\\1&1\end{array}\right)\] have
                  the following property: all infinite periodic products of
                  the two matrices converge to zero, but there exists a
                  nonperiodic product that doesn't. The proof is
                  self-contained and fairly elementary; it uses only
                  elementary facts from the theory of formal languages and
                  from linear algebra. It is not constructive in that we do
                  not exhibit any explicit values of $a$ and $b$ with the
                  stated property; the problem of finding explicit matrices
                  with this property remains open.",
}

@ARTICLE{BT:IPL97,
 author        = "Blondel, Vincent D. and Tsitsiklis, John N.",
 authauthor    = "Blondel, Vincent and Tsitsiklis, John",
 title         = "When is a pair of matrices mortal?",
 journal       = "Inform. Process. Lett.",
 fjournal      = "Information Processing Letters",
 year          = "1997",
 volume        = "63",
 number        = "5",
 pages         = "283--286",
 coden         = "IFPLAT",
 issn          = "0020-0190",
 mrclass       = "15A99 (03D15 03D35 68Q25)",
 mrnumber      = "1475343 (98f:15023)",
 zblnumber     = "1337.68123",
 doi           = "10.1016/S0020-0190(97)00123-3",
 url           = "https://www.sciencedirect.com/science/article/pii/S0020019097001233",
 annote        = "A set of matrices over the integers is said to be $k$-mortal
                  (with $k$ positive integer) if the zero matrix can be
                  expressed as a product of length $k$ of matrices in the
                  set. The set is said to be mortal if it is $k$-mortal for
                  some finite $k$. We show that the problem of deciding
                  whether a pair of $48\times 48$ integer matrices is mortal
                  is undecidable, and that the problem of deciding, for a
                  given $k$, whether a pair of matrices is $k$-mortal is
                  NP-complete.",
}

@ARTICLE{BT:SCL00,
 author        = "Blondel, Vincent D. and Tsitsiklis, John N.",
 authauthor    = "Blondel, Vincent and Tsitsiklis, John",
 title         = "The boundedness of all products of a pair of matrices is
                  undecidable",
 journal       = "Systems Control Lett.",
 fjournal      = "Systems {\&} Control Letters",
 year          = "2000",
 volume        = "41",
 number        = "2",
 pages         = "135--140",
 coden         = "SCLEDC",
 issn          = "0167-6911",
 mrclass       = "93B40 (03D35 15A60 65F30 93D09)",
 mrnumber      = "1831027 (2002b:93054)",
 mrreviewer    = "Nicholas P. Karampetakis",
 zblnumber     = "0985.93042",
 doi           = "10.1016/S0167-6911(00)00049-9",
 url           = "https://www.sciencedirect.com/science/article/pii/S0167691100000499",
 annote        = "We show that the boundedness of the set of all products of a
                  given pair $\Sigma$ of rational matrices is undecidable.
                  Furthermore, we show that the joint (or generalized)
                  spectral radius $\rho(\Sigma)$ is not computable because
                  testing whether $\rho(\Sigma)\le 1$ is an undecidable
                  problem. As a consequence, the robust stability of linear
                  systems under time-varying perturbations is undecidable,
                  and the same is true for the stability of a simple class of
                  hybrid systems. We also discuss some connections with the
                  so-called ``finiteness conjecture''. Our results are based
                  on a simple reduction from the emptiness problem for
                  probabilistic finite automata, which is known to be
                  undecidable.",
}

@ARTICLE{BT:Autom00,
 author        = "Blondel, Vincent D. and Tsitsiklis, John N.",
 authauthor    = "Blondel, Vincent and Tsitsiklis, John",
 title         = "A survey of computational complexity results in systems and
                  control",
 journal       = "Automatica J. IFAC",
 fjournal      = "Automatica. A Journal of IFAC, the International Federation
                  of Automatic Control",
 year          = "2000",
 volume        = "36",
 number        = "9",
 pages         = "1249--1274",
 coden         = "ATCAA9",
 issn          = "0005-1098",
 mrclass       = "93-02 (90C60 93B40 93C65)",
 mrnumber      = "1834719 (2002c:93001)",
 zblnumber     = "0989.93006",
 doi           = "10.1016/S0005-1098(00)00050-9",
 url           = "https://www.sciencedirect.com/science/article/pii/S0005109800000509",
 annote        = "The purpose of this paper is twofold: (a) to provide a
                  tutorial introduction to some key concepts from the theory
                  of computational complexity, highlighting their relevance
                  to systems and control theory, and (b) to survey the
                  relatively recent research activity lying at the interface
                  between these fields. We begin with a brief introduction to
                  models of computation, the concepts of undecidability,
                  polynomial-time algorithms, NP-completeness, and the
                  implications of intractability results. We then survey a
                  number of problems that arise in systems and control
                  theory, some of them classical, some of them related to
                  current research. We discuss them from the point of view of
                  computational complexity and also point out many open
                  problems. In particular, we consider problems related to
                  stability or stabilizability of linear systems with
                  parametric uncertainty, robust control, time-varying linear
                  systems, nonlinear and hybrid systems, and stochastic
                  optimal control.",
}

@ARTICLE{Bochi:LAA03,
 author        = "Bochi, Jairo",
 title         = "Inequalities for numerical invariants of sets of matrices",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2003",
 volume        = "368",
 pages         = "71--81",
 eprint        = "math/0206128",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15A42 (14L24 15A60)",
 mrnumber      = "1983195 (2004f:15034)",
 mrreviewer    = "Ion Zaballa",
 zblnumber     = "1031.15023",
 zblreviewer   = "F. Uhlig (Auburn)",
 doi           = "10.1016/S0024-3795(02)00658-4",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379502006584",
 annote        = "This paper generalizes norm and spectral radius inequalities
                  for single matrices to bounded sets of matrices. Namely,
                  the following statement which follows from the
                  Cayley--Hamilton theorem: if $A \in {\mathbb{C}}^{n,n}$
                  then $\|A^n\|\le (2^n-1) \rho(A) \|A\|^{n-1}$, can be
                  extended to bounded subsets $\Sigma \subset
                  {\mathbb{C}}^{n,n}$:\par \textbf{Theorem.}
                  \emph{$\|\Sigma^n\|\le C {\mathcal{R}}(\Sigma)
                  \|\Sigma\|^{n-1}$ for a constant $C = C(n)$, every bounded
                  subset $\Sigma$ of $n$ by $n$ matrices, and every matrix
                  norm on ${\mathbb{C}}^{n,n}$.}\par Here $\Sigma^n$ denotes
                  the set of all products with $n$ factors from $\Sigma$ and
                  ${\mathcal{R}}(\Sigma) = \lim_{n\to\infty}
                  \|\Sigma^n\|^{^{1/n}}$ in analogy to the spectral radius
                  theorem $\rho(A) = \lim_{n \to \infty} \|A^n\|^{^{1/n}}$
                  for a single matrix $A$. A further result of the paper is
                  as folows:\par \textbf{Theorem.} \emph{The inequality
                  $\mathcal{{R}}(\Sigma) \le C_2 \max_{j \le k}
                  \rho(\Sigma^j)^{^{1/j}}$ is valid with universal constants
                  $C_2 = C_2(n)$ and $k=k(n)$ depending only on the dimension
                  $n$, but not on $\Sigma \subset {\mathbb{C}}^{n,n}$.}\par
                  The latter inequality implies the generalized spectral
                  radius theorem of \emph{M.\,A.~Berger} and
                  \emph{Y.~Wang}~\cite{BerWang:LAA92}. The proofs are
                  topological and use geometric invariant theory, as well as
                  the Cayley--Hamilton theorem.",
}

@UNPUBLISHED{Bochi15,
 author        = "Bochi, Jairo",
 title         = "Another proof of the spectral radius formula",
 year          = "2015",
 url           = "http://www.mat.uc.cl/~jairo.bochi/docs/SRF.pdf",
 note          = "Manuscript",
 annote        = "This note gives an elementary proof of the spectral radius
                  formula in finite dimension, inspired by ideas from ergodic
                  theory.",
}

@ARTICLE{BG:JEP19,
 author        = "Bochi, Jairo and Garibaldi, Eduardo",
 title         = "Extremal norms for fiber-bunched cocycles",
 journal       = "Journal de l'{\'E}cole polytechnique --- Math{\'e}matiques",
 publisher     = "{\'E}cole polytechnique",
 year          = "2019",
 volume        = "6",
 pages         = "947--1004",
 eprinttype    = "arXiv",
 eprint        = "1808.02804",
 issn          = "2429-7100",
 mrclass       = "37H15 (15A60 37D20 37D30 93D30)",
 mrnumber      = "4031530",
 zblnumber     = "1441.37059",
 doi           = "10.5802/jep.109",
 url           = "https://jep.centre-mersenne.org/articles/10.5802/jep.109/",
 annote        = "In traditional Ergodic Optimization, one seeks to maximize
                  Birkhoff averages. The most useful tool in this area is the
                  celebrated Ma{\~n}{\'e} Lemma, in its various forms. In
                  this paper, we prove a non-commutative Ma{\~n}{\'e} Lemma,
                  suited to the problem of maximization of Lyapunov exponents
                  of linear cocycles or, more generally, vector bundle
                  automorphisms. More precisely, we provide conditions that
                  ensure the existence of an extremal norm, that is, a
                  Finsler norm with respect to which no vector can be
                  expanded in a single iterate by a factor bigger than the
                  maximal asymptotic expansion rate. These conditions are
                  essentially irreducibility and sufficiently strong fiber
                  bunching. Therefore we extend the classic concept of
                  Barabanov norm, which is used in the study of the joint
                  spectral radius. We obtain several consequences, including
                  sufficient conditions for the existence of Lyapunov
                  maximizing sets.",
}

@ARTICLE{BochiLas:SAIMJMAA24,
 author        = "Bochi, Jairo and Laskawiec, Piotr",
 title         = "Spectrum Maximizing Products Are Not Generically Unique",
 journal       = "SIAM J. Matrix Anal. Appl.",
 fjournal      = "SIAM Journal on Matrix Analysis and Applications",
 volume        = "45",
 year          = "2024",
 number        = "1",
 pages         = "585--600",
 issn          = "0895-4798,1095-7162",
 mrclass       = "99-06",
 mrnumber      = "4704631",
 zblnumber     = "07805913",
 doi           = "10.1137/23M1550621",
 url           = "https://epubs.siam.org/doi/10.1137/23M1550621",
 eprinttype    = "arXiv",
 eprint        = "2301.12574",
 annote        = "It is widely believed that typical finite families of $d
                  \times d$ matrices admit finite products that attain the
                  joint spectral radius. This conjecture is supported by
                  computational experiments and it naturally leads to the
                  following question: are these spectrum maximizing products
                  typically unique, up to cyclic permutations and powers? We
                  answer this question negatively. As discovered by Horowitz
                  around fifty years ago, there are products of matrices that
                  always have the same spectral radius despite not being
                  cyclic permutations of one another. We show that the
                  simplest Horowitz products can be spectrum maximizing in a
                  robust way; more precisely, we exhibit a small but nonempty
                  open subset of pairs of $2 \times 2$ matrices $(A,B)$ for
                  which the products $A^2 B A B^2$ and $B^2 A B A^2$ are both
                  spectrum maximizing.",
}

@ARTICLE{BochiMor:PLMS15,
 author        = "Bochi, Jairo and Morris, Ian D.",
 authauthor    = "Bochi, Jairo and Morris, Ian",
 title         = "Continuity properties of the lower spectral radius",
 journal       = "Proc. Lond. Math. Soc. (3)",
 fjournal      = "Proceedings of the London Mathematical Society. Third
                  Series",
 year          = "2015",
 volume        = "110",
 number        = "2",
 pages         = "477--509",
 eprinttype    = "arXiv",
 eprint        = "1309.0319",
 issn          = "0024-6115",
 mrclass       = "15A60 (37H15 65F99)",
 mrnumber      = "3335285",
 zblnumber     = "1311.15022",
 doi           = "10.1112/plms/pdu058",
 url           = "https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/plms/pdu058",
 annote        = "The lower spectral radius, or joint spectral subradius, of a
                  set of real $d \times d$ matrices is defined to be the
                  smallest possible exponential growth rate of long products
                  of matrices drawn from that set. The lower spectral radius
                  arises naturally in connection with a number of topics
                  including combinatorics on words, the stability of linear
                  inclusions in control theory, and the study of random
                  Cantor sets. In this article we apply some ideas
                  originating in the study of dominated splittings of linear
                  cocycles over a dynamical system to characterize the points
                  of continuity of the lower spectral radius on the set of
                  all compact sets of invertible $d \times d$ matrices. As an
                  application we exhibit open sets of pairs of $2 \times 2$
                  matrices within which the analogue of the Lagarias--Wang
                  finiteness property for the lower spectral radius fails on
                  a residual set, and discuss some implications of this
                  result for the computation of the lower spectral radius.",
}

@ARTICLE{BochiRams:JMD16,
 author        = "Bochi, Jairo and Rams, Micha{\l}",
 title         = "The entropy of {L}yapunov-optimizing measures of some matrix
                  cocycles",
 journal       = "J. Mod. Dyn.",
 fjournal      = "Journal of Modern Dynamics",
 year          = "2016",
 volume        = "10",
 pages         = "255--286",
 month         = jul,
 eprinttype    = "arXiv",
 eprint        = "1312.6718",
 issn          = "1930-5311",
 mrclass       = "37H15",
 mrnumber      = "3538864",
 mrreviewer    = "Peng Sun",
 zblnumber     = "1346.15030",
 doi           = "10.3934/jmd.2016.10.255",
 url           = "https://www.aimsciences.org/journals/displayArticlesnew.jsp?paperID=12702",
 annote        = "We consider one-step cocycles of $2 \times 2$ matrices, and
                  we are interested in their Lyapunov-optimizing measures,
                  i.e., invariant probability measures that maximize or
                  minimize a Lyapunov exponent. If the cocycle is dominated,
                  that is, the two Lyapunov exponents are uniformly separated
                  along all orbits, then Lyapunov-optimizing measures always
                  exist, and are characterized by their support. Under an
                  additional hypothesis of nonoverlapping between the cones
                  that characterize domination, we prove that the
                  Lyapunov-optimizing measures have zero entropy. This
                  conclusion certainly fails without the domination
                  assumption, even for typical one-step
                  $\mathrm{SL}(2,\mathbb{R})$-cocycles; indeed we show that
                  in the latter case there are measures of positive entropy
                  with zero Lyapunov exponent.",
}

@MISC{Bogdanovic:ArXiv23,
 author        = "Bogdanovi{\'c}, Katarina",
 title         = "Inequalities on the generalized and the joint spectral
                  radius of {H}adamard products",
 howpublished  = "ArXiv.org e-Print archive",
 year          = "2023",
 month         = nov,
 eprinttype    = "arXiv",
 eprint        = "2311.02444",
 doi           = "10.48550/arXiv.2311.02444",
 annote        = "In this article we prove new inequalities for the
                  generalized and the joint spectral radius of bounded sets
                  of positive operators on Banach function and sequence
                  spaces, in particular some inequalities for positive kernel
                  operators on $L^2(X, \mu)$ that generalize, extend and
                  refine several of results obtained recently.",
}

@ARTICLE{BP:LMA23,
 author        = "Katarina Bogdanovi{\'c} and Aljo{\v{s}}a Peperko",
 title         = "Inequalities and equalities on the joint and generalized
                  spectral and essential spectral radius of the {H}adamard
                  geometric mean of bounded sets of positive kernel
                  operators",
 journal       = "Linear Multilinear Algebra",
 fjournal      = "Linear and Multilinear Algebra",
 volume        = "71",
 year          = "2023",
 number        = "17",
 pages         = "2839--2857",
 issn          = "0308-1087,1563-5139",
 mrclass       = "47A10 (15A42 15A60 15B48 47B34 47B65)",
 mrnumber      = "4684437",
 zblnumber     = "07776602",
 doi           = "10.1080/03081087.2022.2121369",
 url           = "https://www.tandfonline.com/doi/full/10.1080/03081087.2022.2121369",
 annote        = "We prove new inequalities and equalities for the generalized
                  and the joint spectral radius (and their essential
                  versions) of Hadamard (Schur) geometric means of bounded
                  sets of positive kernel operators on Banach function
                  spaces. In the case of nonnegative matrices that define
                  operators on Banach sequences we obtain additional results.
                  Our results extend results of several authors that appeared
                  relatively recently.",
}

@MISC{BokPar:ArXiv13,
 author        = "Bokharaie, Vahid and Parsaee, Gholamreza",
 title         = "An application of joint spectral radius in power control
                  problem for wireless communications",
 howpublished  = "ArXiv.org e-Print archive",
 year          = "2013",
 month         = jul,
 eprinttype    = "arXiv",
 eprint        = "1307.0555",
 doi           = "10.48550/arXiv.1307.0555",
 annote        = "Resource management, including power control, is one of the
                  most essential functionalities of any wireless
                  telecommunication system. Various transmitter power-control
                  methods have been developed to deliver a desired quality of
                  service in wireless networks. We consider two of these
                  methods: Distributed Power Control and Distributed
                  Balancing Algorithm schemes. We use the concept of joint
                  spectral radius to come up with conditions for convergence
                  of the transmitted power in these two schemes when the
                  gains on all the communications links are assumed to vary
                  at each time-step.",
}

@ARTICLE{BB:TCS02,
 author        = "Bournez, O. and Branicky, M.",
 title         = "The mortality problem for matrices of low dimensions",
 journal       = "Theory Comput. Syst.",
 fjournal      = "Theory of Computing Systems",
 year          = "2002",
 volume        = "35",
 number        = "4",
 pages         = "433--448",
 coden         = "TCSYFI",
 issn          = "1432-4350",
 mrclass       = "03D35 (03B25 15A30 68Q05)",
 mrnumber      = "1909045 (2004h:03090)",
 mrreviewer    = "Juhani Karhum{\"a}ki",
 zblnumber     = "1016.68038",
 doi           = "10.1007/s00224-002-1010-5",
 url           = "https://link.springer.com/article/10.1007/s00224-002-1010-5",
 annote        = "In this paper we discuss the existence of an algorithm to
                  decide if a given set of $2 \times 2$ matrices is mortal. A
                  set $F=\{A_1,\ldots,A_m\}$ of square matrices is said to be
                  mortal if there exist an integer $k \geq 1$ and some
                  integers $i_1 ,i_2 ,\ldots,i_k\in\{1,\ldots, m\}$ with
                  $A_{i_1} A_{i_2}\cdots A_{i_k} =0$. We survey this problem
                  and propose some new extensions. We prove the problem to be
                  BSS-undecidable for real matrices and Turing-decidable for
                  two rational matrices. We relate the problem for rational
                  matrices to the entry-equivalence problem, to the
                  zero-in-the-corner problem, and to the reachability problem
                  for piecewise-affine functions. Finally, we state some
                  NP-completeness results.",
}

@ARTICLE{BM:JAMS02,
 author        = "Bousch, Thierry and Mairesse, Jean",
 title         = "Asymptotic height optimization for topical {IFS}, {T}etris
                  heaps, and the finiteness conjecture",
 journal       = "J. Amer. Math. Soc.",
 fjournal      = "Journal of the American Mathematical Society",
 year          = "2002",
 volume        = "15",
 number        = "1",
 pages         = "77--111 (electronic)",
 hal_id        = "hal-00165771",
 hal_version   = "v1",
 issn          = "0894-0347",
 mrclass       = "49J27 (37B15 93C65)",
 mrnumber      = "1862798 (2002j:49008)",
 mrreviewer    = "Bart De Schutter",
 zblnumber     = "1057.49007",
 doi           = "10.1090/S0894-0347-01-00378-2",
 url           = "https://www.ams.org/journals/jams/2002-15-01/S0894-0347-01-00378-2/",
 annote        = "Given an Iterated Function System (IFS) of topical maps
                  verifying some conditions, we prove that the asymptotic
                  height optimization problems are equivalent to finding the
                  extrema of a continuous functional, the average height, on
                  some compact space of measures. We give general results to
                  determine these extrema, and then apply them to two
                  concrete problems. First, we give a new proof of the
                  following assertion:\par \textbf{Theorem.} \emph{The
                  densest heaps of two Tetris pieces are sturmian.}\par
                  Second, we construct an explicit counterexample to the
                  Lagarias--Wang finiteness conjecture~\cite{LagWang:LAA95}
                  for the joint spectral radius of a set of matrices.",
}

@ARTICLE{BrayTong:TCS79,
 author        = "Brayton, Robert K. and Tong, Christopher H.",
 authauthor    = "Brayton, Robert and Tong, Christopher",
 title         = "Stability of dynamical systems: a constructive approach",
 journal       = "IEEE Trans. Circuits and Systems",
 fjournal      = "Institute of Electrical and Electronics Engineers.
                  Transactions on Circuits and Systems",
 year          = "1979",
 volume        = "26",
 number        = "4",
 pages         = "224--234",
 coden         = "ICSYBT",
 issn          = "0098-4094",
 mrclass       = "93D05 (15A60 34D20)",
 mrnumber      = "525235 (80d:93053)",
 zblnumber     = "0413.93048",
 doi           = "10.1109/TCS.1979.1084637",
 url           = "https://ieeexplore.ieee.org/document/1084637",
 annote        = "A set $A$ of $n\times n$ complex matrices is stable if for
                  every neighborhood of the origin $U\subset C^{n}$, there
                  exists another neighborhood of the origin $V$, such that
                  for each $M\in A'$ (the set of finite products of matrices
                  in $A$), $MV\subset U$. Matrix and Lyapunov stability are
                  related.\par \textbf{Theorem.} \emph{A set of matrices $A$
                  is stable if and only if there exists a norm, $|\cdot|$,
                  such that for all $M\in A$, and all $z\in C^{n}$,
                  $|Mz|<|z|$.}\par The norm is a \emph{Lyapunov function} for
                  the set $A$. It need not be smooth; using smooth norms to
                  prove stability can be inadequate. A novel central result
                  is a constructive algorithm which can determine the
                  stability of $A$ based on the following assertion:\par
                  \textbf{Theorem.} \emph{$A = \{M_0,M_1,\ldots,M_{m-1}\}$ is
                  a set of $m$ distinct complex matrices. Let $W_0$ be a
                  bounded neighborhood of the origin. Define for $k > 0$,}
                  \[W_k = \textrm{convex hull}~\left[\bigcup_{i=0}^{\infty}
                  M_{k'}^{i}W_{k-1}\right]\] \emph{where $k'=(k- 1) \mod m$.
                  Then $A$ is stable if and only if $W^{*}\equiv
                  \cup_{i=1}^{\infty}W_{i}$ is bounded.}\par The constructive
                  algorithm represents a convex set by its extreme points and
                  uses linear programming to construct the successive $W_k$.
                  Sufficient conditions for the finiteness of constructing
                  $W_k$ from $W_{k-1}$, and for stopping the algorithm when
                  either the set is proved stable or unstable are presented.
                  $A$ is generalized to be any convex set of matrices. A
                  dynamical system of differential equations is stable if a
                  corresponding set of matrices -- associated with the
                  logarithmic norms of the matrices of the linearized
                  equations -- is stable. The concept of the stability of a
                  set of matrices is related to the existence of a matrix
                  norm. Such a norm is an induced matrix norm if and only if
                  the set of matrices is maximally stable (i.e., it cannot be
                  enlarged and remain stable).",
}

@ARTICLE{BrayTong:TCS80,
 author        = "Brayton, Robert K. and Tong, Christopher H.",
 authauthor    = "Brayton, Robert and Tong, Christopher",
 title         = "Constructive stability and asymptotic stability of dynamical
                  systems",
 journal       = "IEEE Trans. Circuits and Systems",
 fjournal      = "Institute of Electrical and Electronics Engineers.
                  Transactions on Circuits and Systems",
 year          = "1980",
 volume        = "27",
 number        = "11",
 pages         = "1121--1130",
 coden         = "ICSYBT",
 issn          = "0098-4094",
 mrclass       = "93D20 (34D20)",
 mrnumber      = "594156 (81j:93044)",
 zblnumber     = "0458.93047",
 doi           = "10.1109/TCS.1980.1084749",
 url           = "https://ieeexplore.ieee.org/document/1084749",
 annote        = "In an earlier paper, the authors presented an algorithm for
                  constructing a Lyapunov function for a dynamical system. In
                  this paper, we present theorems which allow the algorithm
                  to be used in proving the asymptotic stability of dynamical
                  systems, both difference and differential equations. The
                  notion of an asymptotically stable set of matrices is
                  introduced, and is shown to be a sufficient condition for
                  the algorithm's termination in a finite number of steps.
                  The instability stopping criterion is strengthened and the
                  efficiency of the algorithm is improved in a number of
                  ways. We investigate the tightness of our method by
                  applying it to two-dimensional systems for which necessary
                  and sufficient conditions for stability are known.",
}

@INCOLLECTION{Breuillard2022,
 author        = "Breuillard, Emmanuel",
 editor        = "Avila, Artur and Rassias, Michael Th. and Sinai, Yakov",
 title         = "On the joint spectral radius",
 booktitle     = "Analysis at Large: Dedicated to the Life and Work of Jean
                  Bourgain",
 publisher     = "Springer International Publishing",
 address       = "Cham",
 year          = "2022",
 pages         = "1--16",
 eprinttype    = "arXiv",
 eprint        = "2103.09089",
 isbn          = "978-3-031-05331-3",
 doi           = "10.1007/978-3-031-05331-3_1",
 url           = "https://link.springer.com/chapter/10.1007/978-3-031-05331-3_1",
 annote        = "For a bounded subset $S$ of $d\times d$ complex matrices,
                  the Berger--Wang theorem and Bochi's inequality allow to
                  approximate the joint spectral radius of $S$ from below by
                  the spectral radius of a short product of elements from
                  $S$. Our goal is two-fold: we review these results,
                  providing self-contained proofs, and we derive an improved
                  version with explicit bounds that are polynomial in $d$. We
                  also discuss other complete valued fields.",
}

@ARTICLE{BF:AIF21,
 author        = "Breuillard, Emmanuel and Fujiwara, Koji",
 title         = "On the joint spectral radius for isometries of
                  non-positively curved spaces and uniform growth",
 journal       = "Ann. Inst. Fourier (Grenoble)",
 fjournal      = "Universit{\'e} de Grenoble. Annales de l'Institut Fourier",
 year          = "2021",
 volume        = "71",
 number        = "1",
 pages         = "317--391",
 eprinttype    = "arXiv",
 eprint        = "1804.00748",
 issn          = "0373-0956",
 mrclass       = "20F65 (20M05 53C23)",
 mrnumber      = "4275871",
 mrreviewer    = "Henry Hugh Adams",
 zblnumber     = "07362105",
 doi           = "10.5802/aif.3374",
 url           = "http://aif.cedram.org/item?id=AIF_2021__71_1_317_0",
 annote        = "We recast the notion of joint spectral radius in the setting
                  of groups acting by isometries on non-positively curved
                  spaces and give geometric versions of results of
                  Berger--Wang and Bochi valid for $\delta$-hyperbolic spaces
                  and for symmetric spaces of non-compact type. This method
                  produces nice hyperbolic elements in many classical
                  geometric settings. Applications to uniform growth are
                  given, in particular a new proof and a generalization of a
                  theorem of Besson--Courtois--Gallot.",
}

@ARTICLE{BreSert:LMS21,
 author        = "Breuillard, Emmanuel and Sert, Cagri",
 title         = "The joint spectrum",
 journal       = "J. Lond. Math. Soc. (2)",
 fjournal      = "Journal of the London Mathematical Society. Second Series",
 year          = "2021",
 volume        = "103",
 number        = "3",
 pages         = "943--990",
 eprinttype    = "arXiv",
 eprint        = "1809.02404",
 issn          = "0024-6107",
 mrclass       = "15A18 (20G20 22E46 37H15)",
 mrnumber      = "4245827",
 zblnumber     = "07360999",
 doi           = "10.1112/jlms.12397",
 url           = "https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/jlms.12397",
 annote        = "We introduce the notion of \emph{joint spectrum} of a
                  compact set of matrices
                  $S\subset\operatorname{GL}_d(\mathbb{C})$, which is a
                  multi-dimensional generalization of the joint spectral
                  radius. We begin with a thorough study of its properties
                  (under various assumptions: irreducibility,
                  Zariski-density, and domination). Several classical
                  properties of the joint spectral radius are shown to hold
                  in this generalized setting and an analogue of the
                  Lagarias--Wang finiteness conjecture is discussed. Then we
                  relate the joint spectrum to matrix valued random processes
                  and study what points of it can be realized as Lyapunov
                  vectors. We also show how the joint spectrum encodes all
                  word metrics on reductive groups. Several examples are
                  worked out in detail.\par This paper relies extensively on
                  colour figures. Some references to colour may not be
                  meaningful in the printed version, and we refer the reader
                  to the online version which includes the colour figures.",
}

@ARTICLE{BZh:SAIMJMAA00,
 author        = "Br{\"o}ker, Markus and Zhou, Xinlong",
 title         = "Characterization of continuous, four-coefficient scaling
                  functions via matrix spectral radius",
 journal       = "SIAM J. Matrix Anal. Appl.",
 fjournal      = "SIAM Journal on Matrix Analysis and Applications",
 year          = "2000",
 volume        = "22",
 number        = "1",
 pages         = "242--257",
 issn          = "0895-4798",
 mrclass       = "65T60 (15A18 39A10)",
 mrnumber      = "1779727",
 mrreviewer    = "Tomas Sauer",
 zblnumber     = "0964.42017",
 doi           = "10.1137/S0895479897323750",
 url           = "https://epubs.siam.org/doi/10.1137/S0895479897323750",
 annote        = "We characterize the existence of continuous solutions of a
                  four-coefficient dilation equation in terms of the usual
                  spectral radius of a matrix. The criteria for the existence
                  of such a solution can be very quickly examined. As a
                  result we give an affirmative answer to a conjecture raised
                  by Colella and Heil in 1992. Moreover, using our criteria
                  we find the smoothest compactly supported four-coefficient
                  orthogonal scaling function and thus the smoothest
                  compactly supported orthonormal wavelet generated by this
                  scaling function.\par In particular, there is given the
                  following formula for computation of the generalized
                  spectral radius\par \textbf{Lemma 4.} \emph{Suppose $C_{0}$
                  and $C_{1}$ are two $2\times{}2$ matrices. If
                  $\mathop{\mathrm{det}}(C_{0})\le 0$ or
                  $\mathop{\mathrm{det}}(C_{1})\le 0$, then} \[ \rho(C_{0},
                  C_{1}) =\sup_{i+j\ge1,~
                  i,j\ge0}\left(\rho(C^{i}_{0}C^{j}_{1})\right)^{\frac{1}{i+j}},\]
                  \emph{where $\rho(C_{0}, C_{1})$ is the generalized
                  spectral radius of the family of matrices $\{C_{0},
                  C_{1}\}$.}",
}

@ARTICLE{BEN:ETNA94,
 author        = "Bru, Rafael and Elsner, L. and Neumann, M.",
 title         = "Convergence of infinite products of matrices and inner-outer
                  iteration schemes",
 journal       = "Electron. Trans. Numer. Anal.",
 fjournal      = "Electronic Transactions on Numerical Analysis",
 year          = "1994",
 volume        = "2",
 number        = "Dec.",
 pages         = "183--193 (electronic)",
 issn          = "1068-9613",
 mrclass       = "65F10",
 mrnumber      = "1308895 (95i:65046)",
 mrreviewer    = "George D. Poole",
 zblnumber     = "0852.65035",
 url           = "http://etna.mcs.kent.edu/volumes/1993-2000/vol2/abstract.php?vol=2&pages=183-193",
 annote        = "We develop conditions under which a product $\prod^\infty_{i
                  = 0} T_i$ of matrices chosen from a possibly infinite set
                  of matrices $\mathcal{S} = \{T_j \mid j \in J\}$ converges.
                  We obtain the following conditions which are sufficient for
                  the convergence of the product: There exists a vector norm
                  such that all matrices in $\mathcal{S}$ are nonexpansive
                  with respect to this norm and there exists a subsequence
                  $\{i_k\}^\infty_{k = 0}$ of the sequence of the nonnegative
                  integers such that the corresponding sequence of operators
                  $\{T_{i_k}\}^\infty_{k = 0}$ converges to an operator which
                  is paracontracting with respect to this norm. We deduce the
                  continuity of the limit of the product of matrices as a
                  function of the sequences $\{i_k\}^\infty_{k = 0}$. But
                  more importantly, we apply our results to the question of
                  the convergence of inner-outer iteration schemes for
                  solving singular consistent linear systems of equations,
                  where the outer splitting is regular and the inner
                  splitting is weakly regular.",
}

@ARTICLE{BZ:AMPA18,
 author        = "Brundu, M. and Zennaro, M.",
 title         = "Invariant multicones for families of matrices",
 journal       = "Annali di Matematica",
 fjournal      = "Annali di Matematica Pura ed Applicata (1923 -)",
 year          = "2018",
 pages         = "1--44",
 month         = sep,
 day           = "01",
 issn          = "0373-3114",
 mrclass       = "15A18 (15B48 52A30 52B55)",
 mrnumber      = "3927171",
 mrreviewer    = "Ana I. Julio",
 zblnumber     = "1408.15015",
 doi           = "10.1007/s10231-018-0790-4",
 url           = "https://link.springer.com/article/10.1007/s10231-018-0790-4",
 annote        = "In this paper, we investigate sufficient conditions on the
                  structure of the eigenspaces of a given finite family of
                  matrices to assure the existence of an embedded pair of
                  invariant multicones, which are the smallest and the
                  biggest in a suitable and natural sense. Multicones, very
                  similar structures to those known in the literature as
                  1-multicones, are quite natural generalizations of the
                  classical cones. The conditions we find also suggest us a
                  practical computational procedure for the actual
                  construction of such invariant embedded pair.",
}

@ARTICLE{Bui:LAA22,
 author        = "Bui, Vuong",
 title         = "On the joint spectral radius of nonnegative matrices",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2022",
 volume        = "654",
 pages         = "89--101",
 eprinttype    = "arXiv",
 eprint        = "2104.13073",
 issn          = "0024-3795",
 mrclass       = "15A18 (15A60 65F15)",
 mrnumber      = "4481264",
 zblnumber     = "07597857",
 doi           = "10.1016/j.laa.2022.08.029",
 annote        = "We give an effective bound of the joint spectral radius
                  $\rho(\mathcal A)$ for a finite set $\mathcal A$ of
                  nonnegative matrices: For every $n$, \[
                  \sqrt[n]{\Bigl(\frac{V}{UD}\Bigr)^{D} \max_C \max_{i,j\in
                  C} \max_{A_1,\dots,A_n\in\mathcal A}(A_1\dots A_n)_{i,j}}
                  \le \rho(\mathcal A) \le \sqrt[n]{D \max_C \max_{i,j\in C}
                  \max_{A_1,\dots,A_n\in\mathcal A}(A_1\dots A_n)_{i,j}}, \]
                  where $D\times D$ is the dimension of the matrices, $U,V$
                  are respectively the largest entry and the smallest entry
                  over all the positive entries of the matrices in $\mathcal
                  A$, and $C$ is taken over all components in the dependency
                  graph. The dependency graph is a directed graph where the
                  vertices are the dimensions and there is an edge from $i$
                  to $j$ if and only if $A_{i,j}\ne 0$ for some matrix
                  $A\in\mathcal A$.\par Furthermore, a bound on the norm is
                  also given: There exists a nonnegative integer $r$ so that
                  for every $n$, \[\mathop{\mathrm{const}} n^r{\rho(\mathcal
                  A)}^n \le \max_{A_1,\dots,A_n\in\mathcal A} \|A_1\dots
                  A_n\| \le \mathop{\mathrm{const}} n^r{\rho(\mathcal A)}^n.
                  \] Corollaries of the approach include a simple proof for
                  the joint spectral theorem for finite sets of nonnegative
                  matrices and the convergence rate of some sequences. The
                  method in use is mostly based on Fekete's lemma.",
}

@ARTICLE{CACJ:FI21,
 author        = "Catalano, Costanza and Azfar, Umer and Charlier, Ludovic and
                  Jungers, Rapha{\"e}l M.",
 authauthor    = "Catalano, Costanza and Azfar, Umer and Charlier, Ludovic and
                  Jungers, Rapha{\"e}l",
 title         = "A Linear Bound on the $k$-rendezvous Time for Primitive Sets
                  of {NZ} Matrices",
 journal       = "Fund. Inform.",
 fjournal      = "Fundamenta Informaticae",
 year          = "2021",
 volume        = "180",
 number        = "4",
 pages         = "289--314",
 eprinttype    = "arXiv",
 eprint        = "1903.10421",
 issn          = "0169-2968",
 mrclass       = "68Q45 (15A30)",
 mrnumber      = "4288573",
 zblnumber     = "1497.68253",
 doi           = "10.3233/FI-2021-2043",
 url           = "https://content.iospress.com/articles/fundamenta-informaticae/fi2043",
 annote        = "A set of nonnegative matrices is called primitive if there
                  exists a product of these matrices that is entrywise
                  positive. Motivated by recent results relating
                  synchronizing automata and primitive sets, we study the
                  length of the shortest product of a primitive set having a
                  column or a row with $k$ positive entries, called its
                  $k$-rendezvous time ($k$-RT), in the case of sets of
                  matrices having no zero rows and no zero columns. We prove
                  that the $k$-RT is at most linear w.r.t. the matrix size
                  $n$ for small $k$, while the problem is still open for
                  synchronizing automata. We provide two upper bounds on the
                  $k$-RT: the second is an improvement of the first one,
                  although the latter can be written in closed form. We then
                  report numerical results comparing our upper bounds on the
                  k-RT with heuristic approximation methods.",
}

@INCOLLECTION{CosJun:LNCS18,
 author        = "Catalano, Costanza and Jungers, Rapha{\"e}l M.",
 authauthor    = "Costanza Catalano and Rapha{\"e}l Jungers",
 title         = "The synchronizing probability function for primitive sets of
                  matrices",
 booktitle     = "Developments in language theory",
 series        = "Lecture Notes in Comput. Sci.",
 publisher     = "Springer, Cham",
 year          = "2018",
 volume        = "11088",
 pages         = "194--205",
 mrclass       = "68Q45 (68Q19)",
 mrnumber      = "3855942",
 zblnumber     = "06983378",
 doi           = "10.1007/978-3-319-98654-8_16",
 url           = "https://link.springer.com/chapter/10.1007\%2F978-3-319-98654-8_16",
 annote        = "Motivated by recent results relating synchronizing automata
                  and primitive sets, we tackle the synchronization process
                  and the related longstanding {\v{C}}ern{\'y} conjecture by
                  studying the primitivity phenomenon for sets of nonnegative
                  matrices having neither zero-rows nor zero-columns. We
                  formulate the primitivity process in the setting of a
                  two-player probabilistic game and we make use of convex
                  optimization techniques to describe its behavior. We report
                  numerical results and supported by them we state a
                  conjecture that, if true, would imply an upper bound of
                  $n(n-1)$ on the reset threshold of a certain class of
                  automata.",
}

@ARTICLE{CosJun:IJFCS20,
 author        = "Catalano, Costanza and Jungers, Rapha{\"e}l M.",
 authauthor    = "Costanza Catalano and Rapha{\"e}l Jungers",
 title         = "The synchronizing probability function for primitive sets of
                  matrices",
 journal       = "Internat. J. Found. Comput. Sci.",
 fjournal      = "International Journal of Foundations of Computer Science",
 year          = "2020",
 volume        = "31",
 number        = "6",
 pages         = "777--803",
 issn          = "0129-0541",
 mrclass       = "Prelim",
 mrnumber      = "4174088",
 zblnumber     = "1458.68093",
 doi           = "10.1142/S0129054120410051",
 url           = "https://www.worldscientific.com/doi/abs/10.1142/S0129054120410051",
 annote        = "Motivated by recent results relating synchronizing DFAs and
                  primitive sets, we tackle the synchronization process and
                  the related longstanding {\v{C}}ern{\'y} conjecture by
                  studying the primitivity phenomenon for sets of nonnegative
                  matrices having neither zero-rows nor zero-columns. We
                  formulate the primitivity process in the setting of a
                  two-player probabilistic game and we make use of convex
                  optimization techniques to describe its behavior. We
                  develop a tool for approximating and upper bounding the
                  exponent of any primitive set and supported by numerical
                  results we state a conjecture that, if true, would imply a
                  quadratic upper bound on the reset threshold of a new class
                  of automata.",
}

@PHDTHESIS{ChangPhD12,
 author        = "Chang, Chia-Tche",
 authauthor    = "Chang, Chia-Tche",
 title         = "Heuristic optimization methods for three matrix problem",
 school        = "Universit{\'e} catholique de Louvain, Louvain School of
                  Engineering",
 address       = "Louvain-la-Neuve, Belgium",
 year          = "2012",
 url           = "https://perso.uclouvain.be/chia-tche.chang/research.php",
 annote        = "Optimization is a major field in applied mathematics. Many
                  applications involve the search of the best solution to a
                  problem according to some criterion. Depending on the
                  considered optimization problem, finding the best solution
                  is not always possible in a reasonable amount of time.
                  Heuristic algorithms are often used when the problem is too
                  difficult to solve exactly. These methods are used to speed
                  up the search for a good solution but they do not guarantee
                  that an optimal solution will be found. In this thesis, we
                  explore such heuristic approaches for three different
                  matrix problems. First, we study the minimum-volume
                  bounding box problem, which consists in finding the
                  smallest rectangular parallelepiped enclosing a given set
                  of points in the three-dimensional space. This problem
                  appears for example in collision detection, which is a very
                  important issue in computational geometry and computer
                  graphics. In these applications, a solution has to be
                  determined in a very short amount of time. We propose a new
                  hybrid algorithm able to approximate optimal bounding boxes
                  at a low computational cost. In particular, it is several
                  orders of magnitude faster than the only currently known
                  exact algorithm. Second, we investigate the subset
                  selection problem. Given a large set of features, we want
                  to choose a small subset containing the most relevant
                  features while removing the redundant ones. This problem
                  has applications in data mining since this can be seen as a
                  dimensionality reduction problem. We develop several
                  windowed algorithms that tackle the subset selection
                  problem for the maximum-volume criterion, which is NP-hard.
                  Finally, we address the topic of the approximation of the
                  joint spectral radius. This quantity characterizes the
                  growth rate of product of matrices and is NP-hard to
                  approximate. The joint spectral radius appears in many
                  fields, including system theory, graph theory,
                  combinatorics, language theory\ldots{} We present an
                  experimental study of existing approaches and propose a new
                  genetic-based algorithm that is able to find bounds on the
                  joint spectral radius in a short amount of time.",
}

@INPROCEEDINGS{CB:IFACWC11,
 author        = "Chang, Chia-Tche and Blondel, Vincent",
 title         = "Approximating the Joint Spectral Radius Using a Genetic
                  Algorithm Framework",
 booktitle     = "Proceedings of the 18th IFAC World Congress",
 organization  = "IFAC",
 year          = "2011",
 volume        = "18, part 1",
 pages         = "8681--8686",
 doi           = "10.3182/20110828-6-IT-1002.01412",
 annote        = "The joint spectral radius of a set of matrices is a measure
                  of the maximal asymptotic growth rate of products of
                  matrices in the set. This quantity appears in many
                  applications but is known to be difficult to approximate.
                  Several approaches to approximate the joint spectral radius
                  involve the construction of long products of matrices, or
                  the construction of an appropriate extremal matrix norm. In
                  this article we present a brief overview of several recent
                  approximation algorithms and introduce a genetic algorithm
                  approximation method. This new method does not give any
                  accuracy guarantees but is quite fast in comparison to
                  other techniques. The performances of the different methods
                  are compared and are illustrated on some benchmark
                  examples. Our results show that, for large sets of matrices
                  or matrices of large dimension, our genetic algorithm may
                  provide better estimates or estimates for situations where
                  these are simply too expensive to compute with other
                  methods. As an illustration of this we compute in less than
                  a minute a bound on the capacity of a code avoiding a given
                  forbidden pattern that improves the bound currently
                  reported in the literature.",
}

@ARTICLE{CB:NA13,
 author        = "Chang, Chia-Tche and Blondel, Vincent D.",
 authauthor    = "Chang, Chia-Tche and Blondel, Vincent",
 title         = "An experimental study of approximation algorithms for the
                  joint spectral radius",
 journal       = "Numer. Algorithms",
 fjournal      = "Numerical Algorithms",
 year          = "2013",
 volume        = "64",
 number        = "1",
 pages         = "181--202",
 issn          = "1017-1398",
 mrclass       = "Preliminary Data",
 mrnumber      = "3090842",
 zblnumber     = "1276.65021",
 doi           = "10.1007/s11075-012-9661-z",
 url           = "https://link.springer.com/article/10.1007/s11075-012-9661-z",
 annote        = "We describe several approximation algorithms for the joint
                  spectral radius and compare their performance on a large
                  number of test cases. The joint spectral radius of a set
                  $\Sigma$ of $n\times n$ matrices is the maximal asymptotic
                  growth rate that can be obtained by forming products of
                  matrices from $\Sigma$. This quantity is NP-hard to compute
                  and appears in many areas, including in system theory,
                  combinatorics and information theory. A dozen algorithms
                  have been proposed this last decade for approximating the
                  joint spectral radius but little is known about their
                  practical efficiency. We overview these approximation
                  algorithms and classify them in three categories:
                  approximation obtained by examining long products, by
                  building a specific matrix norm, and by using
                  optimization-based techniques. All these algorithms are now
                  implemented in a (freely available) MATLAB toolbox that was
                  released in 2011. This toolbox allows us to present a
                  comparison of the approximations obtained on a large number
                  of test cases as well as on sets of matrices taken from the
                  literature. Finally, in our comparison we include a method,
                  available in the toolbox, that combines different existing
                  algorithms and that is the toolbox's default method. This
                  default method was able to find optimal products for all
                  test cases of dimension less than four.",
}

@ARTICLE{Charina:ACHA13,
 author        = "Charina, Maria",
 title         = "Finiteness conjecture and subdivision",
 journal       = "Appl. Comput. Harmon. Anal.",
 fjournal      = "Applied and Computational Harmonic Analysis",
 year          = "2014",
 volume        = "36",
 number        = "3",
 pages         = "522--526",
 issn          = "1063-5203",
 mrclass       = "15A60",
 mrnumber      = "3175093",
 zblnumber     = "1311.15023",
 doi           = "10.1016/j.acha.2013.09.002",
 url           = "https://www.sciencedirect.com/science/article/pii/S1063520313000894",
 annote        = "The finiteness conjecture by J.\,C.~Lagarias and
                  Y.~Wang~\cite{LagWang:LAA95} states that the joint spectral
                  radius of a finite set of square matrices is attained on
                  some finite product of such matrices. This conjecture is
                  known to be false in general. Nevertheless, we show that
                  this conjecture is true for a big class of finite sets of
                  square matrices used for the smoothness analysis of scalar
                  univariate subdivision schemes with finite masks.",
}

@INCOLLECTION{CCS:AT04,
 author        = "Charina, Maria and Conti, Costanza and Sauer, Thomas",
 title         = "{$L_p$}-convergence of subdivision schemes: joint spectral
                  radius versus restricted spectral radius",
 booktitle     = "Approximation theory {XI}: {G}atlinburg 2004",
 series        = "Mod. Methods Math.",
 publisher     = "Nashboro Press, Brentwood, TN",
 year          = "2005",
 pages         = "129--150",
 mrclass       = "42C40 (41A30)",
 mrnumber      = "2126678 (2005k:42094)",
 mrreviewer    = "Palle E. T. Jorgensen",
 zblnumber     = "1074.65021",
 annote        = "We compare two approaches for investigating the
                  $L_p$-convergence of multivariate scalar and vector
                  subdivision schemes. The first approach is based on the
                  so-called joint spectral radius, the second one on what we
                  call the restricted spectral radius, a quantity that can be
                  used to characterize the restricted contractivity of the
                  corresponding difference subdivision schemes. We show that
                  the two approaches are not only equivalent, but that in
                  fact the joint spectral radius and the restricted spectral
                  radius are equal for $1\le p\le\infty$. One of the
                  advantages of working with the restricted spectral radius
                  is that the restricted $p$-norms can be computed by means
                  of classical optimization methods, the restricted
                  $\infty$-norm even by means of linear programming. This
                  also allows for an estimation of the restricted spectral
                  radii.",
}

@ARTICLE{ChM:LAA69,
 author        = "Chazan, D. and Miranker, W.",
 title         = "Chaotic relaxation",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "1969",
 volume        = "2",
 pages         = "199--222",
 mrclass       = "65.35",
 mrnumber      = "0251888 (40 \#5114)",
 mrreviewer    = "J. D. P. Donnelly",
 zblnumber     = "0225.65043",
 doi           = "10.1016/0024-3795(69)90028-7",
 url           = "https://www.sciencedirect.com/science/article/pii/0024379569900287",
 annote        = "In this paper we consider relaxation methods for solving
                  linear systems of equations. These methods are suited for
                  execution on a parallel system of processors. They have the
                  feature of allowing a minimal amount of communication of
                  computational status between the computers, so that the
                  relaxation process, while taking on a chaotic appearance,
                  reduces programming and processor time of a bookkeeping
                  nature. We give a precise characterization of chaotic
                  relaxation, some examples of divergence, and conditions
                  guaranteeing convergence.",
}

@ARTICLE{CheMam:BASRM04,
 author        = "Cheban, D. and Mammana, C.",
 title         = "Asymptotic stability of autonomous and non-autonomous
                  discrete linear inclusions",
 journal       = "Bul. Acad. {\c{S}}tiin{\c{t}}e Repub. Mold. Mat.",
 fjournal      = "Buletinul Academiei de {\c{S}}tiin{\c{t}}e a Republicii
                  Moldova. Matematica. Izvestiya Akademii Nauk Respubliki
                  Moldova. Matematika",
 year          = "2004",
 number        = "3",
 pages         = "41--52",
 issn          = "1024-7696",
 mrclass       = "39A11 (34A60 37B25)",
 mrnumber      = "2148008 (2006b:39015)",
 mrreviewer    = "Sung Kyu Choi",
 zblnumber     = "1080.39006",
 zblreviewer   = "Jean Mawhin (Louvain-La-Neuve)",
 annote        = "This paper deals with the absolute asymptotic stability of
                  discrete linear inclusions of the form $x_{t+1}\in F(x_t)$,
                  where $F(x)=\{A_1 x,A_2 x,\ldots,A_m x\}$ and the $A_i$ are
                  linear bounded operators acting on a Banach space. The
                  relations between absolute asymptotic stability, uniform
                  asymptotic stability and uniform exponential stability are
                  analyzed, and they are proved to be equivalent for compact
                  inclusions.",
}

@ARTICLE{ChenZhou:LAA00,
 author        = "Chen, Quande and Zhou, Xinlong",
 title         = "Characterization of joint spectral radius via trace",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2000",
 volume        = "315",
 number        = "1-3",
 pages         = "175--188",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15A18 (65F15)",
 mrnumber      = "1774967 (2002h:15015)",
 mrreviewer    = "Zden{\v{e}}k Dost{\'a}l",
 zblnumber     = "0970.15020",
 zblreviewer   = "F. Uhlig (Auburn)",
 doi           = "10.1016/S0024-3795(00)00149-X",
 url           = "https://www.sciencedirect.com/science/article/pii/S002437950000149X",
 annote        = "The joint spectral radius for a bounded collection of the,
                  square matrices with complex entries and of the same size
                  is characterized by the trace of matrices. This
                  characterization allows us to give some estimates
                  concerning the computation of the joint spectral
                  radius.\par For a bounded set $\Sigma$ of square matrices
                  the joint spectral radius $\rho(\Sigma)$ is defined as the
                  limit superior of the supremum over the set $\mathcal{L}_n$
                  of all matrix products of length $n$ from the collection
                  $\Sigma$ of the $n^{th}$ root of the product matrix norm,
                  while the generalized spectral radius $\widetilde
                  \rho(\Sigma)$ is the same expression involving the spectral
                  radii of the matrix products instead.\par Inequalities such
                  as $\sup \rho(A)^{1/n} \leq \widetilde \rho(\Sigma) \leq
                  \rho(\Sigma) \leq \sup \|A\|^{1/n}$ are known. For a finite
                  set $\Sigma$ of matrices this paper establishes that
                  $\rho(\Sigma) = \limsup\limits_{n \to \infty} \sup
                  \limits_{A \in \mathcal{{L}}_n} |\textrm{trace}(A)|^{1/n}$
                  and gives estimates for the joint spectral radius for sets
                  generated by two matrices in terms of the trace. These
                  concepts apply to cascade algorithms for wavelets with
                  compact support.",
}

@INPROCEEDINGS{CUH:ECC2020,
 author        = "Chenavier, Cyrille and Ushirobira, Rosane and Hetel,
                  Laurentiu",
 title         = "Normal forms of matrix words for stability analysis of
                  discrete-time switched linear systems",
 booktitle     = "2020 European Control Conference (ECC), 12-15 May",
 publisher     = "IEEE",
 address       = "Saint Petersburg, Russia",
 year          = "2020",
 pages         = "1842--1846",
 month         = may,
 hal_id        = "hal-02069712",
 hal_version   = "v3",
 doi           = "10.23919/ECC51009.2020.9143862",
 url           = "https://ieeexplore.ieee.org/document/9143862",
 annote        = "In this paper, we propose a new method for investigating the
                  stability of discrete-time switched linear systems by means
                  of linear algebra techniques. Exponential stability of such
                  systems is equivalent to the existence of solutions of
                  linear matrix inequalities indexed by matrix words, i.e.,
                  products of matrices of the sub-systems). Our method
                  consists in using the link between this characterization of
                  exponential stability and linear dependency among matrix
                  words. In particular, we introduce a criterion to reduce
                  drastically the number of linear matrix inequalities, by
                  removing redundant ones. This is achieved by eliminating
                  matrix words that depend on the others. From this
                  criterion, we also relate exponential stability to
                  quadratic stability of another switched system. An example
                  is given to illustrate our methods.",
}

@ARTICLE{CHJ:SIAMJCO15,
 author        = "Chevalier, Pierre-Yves and Hendrickx, Julien M. and Jungers,
                  Rapha{\"e}l M.",
 authauthor    = "Chevalier, Pierre-Yves and Hendrickx, Julien and Jungers,
                  Rapha{\"e}l",
 title         = "Efficient Algorithms for the Consensus Decision Problem",
 journal       = "SIAM J. Control Optim.",
 fjournal      = "SIAM Journal on Control and Optimization",
 year          = "2015",
 volume        = "53",
 number        = "5",
 pages         = "3104--3119",
 eprinttype    = "arXiv",
 eprint        = "1409.6505",
 issn          = "0363-0129",
 mrclass       = "Preliminary Data",
 mrnumber      = "3403131",
 zblnumber     = "1320.93073",
 doi           = "10.1137/140988024",
 url           = "https://epubs.siam.org/doi/10.1137/1409880244",
 annote        = "We address the problem of determining if a discrete time
                  switched consensus system converges for any switching
                  sequence and that of determining if it converges for at
                  least one switching sequence. For these two problems, we
                  provide necessary and sufficient conditions that can be
                  checked in singly exponential time. As a side result, we
                  prove the existence of a polynomial time algorithm for the
                  first problem when the system switches between only two
                  subsystems whose corresponding graphs are undirected, a
                  problem that had been suggested to be NP-hard by Blondel
                  and Olshevsky.",
}

@ARTICLE{CHJ:SCL17,
 author        = "Chevalier, Pierre-Yves and Hendrickx, Julien M. and Jungers,
                  Rapha{\"e}l M.",
 authauthor    = "Chevalier, Pierre-Yves and Hendrickx, Julien and Jungers,
                  Rapha{\"e}l",
 title         = "Tight bound for deciding convergence of consensus systems",
 journal       = "Systems Control Lett.",
 fjournal      = "Systems {\&} Control Letters",
 year          = "2017",
 volume        = "105",
 pages         = "78--83",
 eprinttype    = "arXiv",
 eprint        = "1601.04975",
 issn          = "0167-6911",
 mrclass       = "15A60 (93D20)",
 mrnumber      = "3668032",
 zblnumber     = "1372.93009",
 doi           = "10.1016/j.sysconle.2017.05.001",
 url           = "https://www.sciencedirect.com/science/article/pii/S0167691117300877",
 annote        = "We analyze the asymptotic convergence of all infinite
                  products of matrices taken in a given finite set by looking
                  only at \emph{finite} or \emph{periodic} products. It is
                  known that when the matrices of the set have a common
                  nonincreasing polyhedral norm, all infinite products
                  converge to zero if and only if all infinite
                  \emph{periodic} products with periods smaller than a
                  certain value converge to zero. Moreover, bounds on that
                  value are available~\cite{LagWang:LAA95}.\par We provide a
                  stronger bound that holds for both polyhedral norms and
                  polyhedral seminorms. In the latter case, the matrix
                  products do not necessarily converge to 0, but all
                  trajectories of the associated system converge to a common
                  invariant subspace. We prove that our bound is tight for
                  all seminorms.\par Our work is motivated by problems in
                  \emph{consensus systems}, where the matrices are
                  \emph{stochastic} (nonnegative with rows summing to one),
                  and hence always share a same common nonincreasing
                  polyhedral seminorm. In that case, we also improve existing
                  results.",
}

@ARTICLE{CGM:JDE15,
 author        = "Chitour, Y. and Gaye, M. and Mason, P.",
 authauthor    = "Yacine Chitour and Moussa Gaye and Paolo Mason",
 title         = "Geometric and asymptotic properties associated with linear
                  switched systems",
 journal       = "J. Differential Equations",
 fjournal      = "Journal of Differential Equations",
 year          = "2015",
 volume        = "259",
 number        = "11",
 pages         = "5582--5616",
 eprinttype    = "arXiv",
 eprint        = "1409.4524",
 hal_id        = "hal-01064241",
 hal_version   = "v2",
 issn          = "0022-0396",
 mrclass       = "93D09 (34A38 34D08 37D25 49K15 49K30)",
 mrnumber      = "3397301",
 zblnumber     = "1334.34035",
 zblreviewer   = "Christian P{\"o}tzsche",
 doi           = "10.1016/j.jde.2015.07.001",
 url           = "https://www.sciencedirect.com/science/article/pii/S0022039615003526",
 annote        = "Consider continuous-time linear switched systems on
                  $\mathbb{R}^n$ associated with compact convex sets of
                  matrices. When the system is irreducible and the largest
                  Lyapunov exponent is equal to zero, there always exists a
                  Barabanov norm (i.e. a norm which is non increasing along
                  trajectories of the linear switched system together with
                  extremal trajectories starting at every point, that is
                  trajectories of the linear switched system with constant
                  norm). This paper deals with two sets of issues: $(a)$
                  properties of Barabanov norms such as uniqueness up to
                  homogeneity and strict convexity; $(b)$ asymptotic
                  behaviour of the extremal solutions of the linear switched
                  system. Regarding Issue $(a)$, we provide partial answers
                  and propose four open problems motivated by appropriate
                  examples. As for Issue $(b)$, we establish, when $n=3$, a
                  Poincar{\'e}-Bendixson theorem under a regularity
                  assumption on the set of matrices defining the system.
                  Moreover, we revisit the noteworthy result of \emph{N.E.
                  Barabanov} [Dokl. Akad. Nauk 334(2), 154--155 (1994)]
                  dealing with the linear switched system on $\mathbb{R}^3$
                  associated with a pair of Hurwitz matrices $\{A,A+bc^T\}$.
                  We first point out a fatal gap in Barabanov's argument in
                  connection with geometric features associated with a
                  Barabanov norm. We then provide partial answers relative to
                  the asymptotic behavior of this linear switched system.",
}

@ARTICLE{CGSP:NAHS21,
 author        = "Chitour, Yacine and Guglielmi, Nicola and Protasov, Vladimir
                  {\relax{}Yu}. and Sigalotti, Mario",
 authauthor    = "Yacine Chitour and Nicola Guglielmi and Mario Sigalotti and
                  Vladimir Protasov",
 title         = "Switching systems with dwell time: {C}omputing the maximal
                  {L}yapunov exponent",
 journal       = "Nonlinear Anal. Hybrid Syst.",
 fjournal      = "Nonlinear Analysis. Hybrid Systems",
 year          = "2021",
 volume        = "40",
 pages         = "101021",
 eprinttype    = "arXiv",
 eprint        = "1912.10214",
 hal_id        = "hal-02423619",
 hal_version   = "v2",
 issn          = "1751-570X",
 mrclass       = "37B25 (15A60 34D08 37M25 39A30)",
 mrnumber      = "4219337",
 zblnumber     = "1485.92078",
 doi           = "10.1016/j.nahs.2021.101021",
 url           = "https://www.sciencedirect.com/science/article/abs/pii/S1751570X2100011X",
 annote        = "We study asymptotic stability of continuous-time systems
                  with mode-dependent guaranteed dwell time. These systems
                  are reformulated as special cases of a general class of
                  mixed (discrete-continuous) linear switching systems on
                  graphs, in which some modes correspond to discrete actions
                  and some others correspond to continuous-time evolutions.
                  Each discrete action has its own positive weight which
                  accounts for its time-duration. We develop a theory of
                  stability for the mixed systems; in particular, we prove
                  the existence of an invariant Lyapunov norm for mixed
                  systems on graphs and study its structure in various cases,
                  including discrete-time systems for which discrete actions
                  have inhomogeneous time durations. This allows us to adapt
                  recent methods for the joint spectral radius computation
                  (Gripenberg's algorithm and the Invariant Polytope
                  Algorithm) to compute the Lyapunov exponent of mixed
                  systems on graphs.",
}

@ARTICLE{CMS:SCL12,
 author        = "Chitour, Yacine and Mason, Paolo and Sigalotti, Mario",
 authauthor    = "Chitour, Yacine and Mason, Paolo and Sigalotti, Mario",
 title         = "On the marginal instability of linear switched systems",
 journal       = "Systems Control Lett.",
 fjournal      = "Systems {\&} Control Letters",
 year          = "2012",
 volume        = "61",
 number        = "6",
 pages         = "747--757",
 coden         = "SCLEDC",
 hal_id        = "inria-00578133",
 hal_version   = "v4",
 issn          = "0167-6911",
 mrclass       = "93D05 (34D05 34D08 93C65)",
 mrnumber      = "2929512",
 zblnumber     = "1250.93112",
 doi           = "10.1016/j.sysconle.2012.04.005",
 url           = "https://hal.inria.fr/inria-00578133",
 annote        = "Stability properties for continuous-time linear switched
                  systems are at first determined by the (largest) Lyapunov
                  exponent associated with the system, which is the analogue
                  of the joint spectral radius for the discrete-time case.
                  The purpose of this paper is to provide a characterization
                  of marginally unstable systems, i.e., systems for which the
                  Lyapunov exponent is equal to zero and there exists an
                  unbounded trajectory, and to analyze the asymptotic
                  behavior of their trajectories. Our main contribution
                  consists in pointing out a resonance phenomenon associated
                  with marginal instability. In the course of our study, we
                  derive an upper bound of the state at time $t$, which is
                  polynomial in $t$ and whose degree is computed from the
                  resonance structure of the system. We also derive analogous
                  results for discrete-time linear switched systems.",
}

@ARTICLE{CMS:MRB22,
 author        = "Chitour, Yacine and Mason, Paolo and Sigalotti, Mario",
 title         = "Characterization of linear switched systems admitting a
                  {B}arabanov norm",
 journal       = "Math. Rep. (Bucur.)",
 fjournal      = "Mathematical Reports (Bucure{\c{s}}ti)",
 year          = "2022",
 volume        = "24(74)",
 number        = "1--2",
 pages         = "125--137",
 issn          = "1582-3067",
 hal_id        = "hal-03468638",
 hal_version   = "v1",
 mrclass       = "93D09 (34D08 93C30)",
 mrnumber      = "4431496",
 url           = "http://imar.ro/journals/Mathematical_Reports/Pdfs/2022/1-2/7.pdf",
 annote        = "In this paper we recall some general properties of extremal
                  and Barabanov norms and we give a necessary and sufficient
                  condition for a finite-dimensional continuoustime linear
                  switched system to admit a Barabanov norm.",
}

@ARTICLE{CMMS:EJP23,
 author        = "Chitour, Yacine and Mazanti, Guilherme and Monmarch{\'e},
                  Pierre and Sigalotti, Mario ",
 title         = "On the gap between deterministic and probabilistic
                  {L}yapunov exponents for continuous-time linear systems",
 journal       = "Electronic Journal of Probability",
 publisher     = "Institute of Mathematical Statistics and Bernoulli Society",
 year          = "2023",
 volume        = "28",
 pages         = "1--39",
 eprinttype    = "arXiv",
 eprint        = "2112.07005",
 hal_id        = "hal-03478271",
 hal_version   = "v2",
 mrclass       = "60J25 (34D08)",
 mrnumber      = "4560991",
 zblnumber     = "07707092",
 doi           = "10.1214/23-EJP932",
 url           = "https://doi.org/10.1214/23-EJP932",
 annote        = "Consider a non-autonomous continuous-time linear system in
                  which the time-dependent matrix determining the dynamics is
                  piecewise constant and takes finitely many values $A_1,
                  \dotsc, A_N$. This paper studies the equality cases between
                  the maximal Lyapunov exponent associated with the set of
                  matrices $\{A_1, \dotsc, A_N\}$, on the one hand, and the
                  corresponding ones for piecewise deterministic Markov
                  processes with modes $A_1, \dotsc, A_N$, on the other hand.
                  A fundamental step in this study consists in establishing a
                  result of independent interest, namely, that any sequence
                  of Markov processes associated with the matrices
                  $A_1,\dotsc, A_N$ converges, up to extracting a
                  subsequence, to a Markov process associated with a suitable
                  convex combination of those matrices.",
}

@ARTICLE{CMS:LAA21,
 author        = "Chitour, Yacine and Mazanti, Guilherme and Sigalotti,
                  Mario",
 title         = "On the gap between deterministic and probabilistic joint
                  spectral radii for discrete-time linear systems",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2021",
 volume        = "613",
 pages         = "24--45",
 eprinttype    = "arXiv",
 eprint        = "1812.08399",
 hal_id        = "hal-01961003",
 hal_version   = "v4",
 issn          = "0024-3795",
 mrclass       = "93B25 (15B48 37H15 60J20)",
 mrnumber      = "4192243",
 zblnumber     = "07312092",
 doi           = "10.1016/j.laa.2020.12.013",
 url           = "https://www.sciencedirect.com/science/article/abs/pii/S002437952030567X",
 annote        = "Given a discrete-time linear switched system
                  $\Sigma(\mathcal A)$ associated with a finite set $\mathcal
                  A$ of matrices, we consider the measures of its asymptotic
                  behavior given by, on the one hand, its deterministic joint
                  spectral radius $\rho_{\mathrm d}(\mathcal A)$ and, on the
                  other hand, its probabilistic joint spectral radii
                  $\rho_{\mathrm p}(\nu,P,\mathcal A)$ for Markov random
                  switching signals with transition matrix $P$ and a
                  corresponding invariant probability $\nu$. Note that
                  $\rho_{\mathrm d}(\mathcal A)$ is larger than or equal to
                  $\rho_{\mathrm p}(\nu,P,\mathcal A)$ for every pair $(\nu,
                  P)$. In this paper, we investigate the cases of equality of
                  $\rho_{\mathrm d}(\mathcal A)$ with either a single
                  $\rho_{\mathrm p}(\nu,P,\mathcal A)$ or with the supremum
                  of $\rho_{\mathrm p}(\nu,P,\mathcal A)$ over $(\nu,P)$ and
                  we aim at characterizing the sets $\mathcal A$ for which
                  such equalities may occur.",
}

@PHDTHESIS{CiconePhD11,
 author        = "Cicone, Antonio",
 title         = "Spectral Properties of Families of Matrices",
 school        = "Universit{\`a} degli Studi dell'Aquila, Facolt{\`a} di
                  Scienze, Dipartimento di Matematica Pura ed Applicata",
 address       = "L'Aquila",
 year          = "2011",
 url           = "https://users.math.msu.edu/users/cicone/papers/antoniociconethesis.pdf",
 annote        = "During the past few decades there has been, an increasing
                  interest in studying the behavior of long products
                  generated using matrices of a given generic family and in
                  particular in analysing the maximal growth rate of these
                  products. This study can be done considering the
                  generalization of the spectral radius of a matrix to the
                  case of a family of matrices, which is called \emph{joint
                  spectral radius} or simply \emph{spectral radius} and it
                  was first introduced by Rota and Strang in the three pages
                  paper ``A note on the joint spectral
                  radius''~\cite{RotaStr:IM60}. To be precise this
                  generalization can be formulated in many different ways,
                  but for the families of matrices which are common in
                  applications, i.e. bounded and finite, all the possible
                  generalizations coincide with each other in a unique value
                  that is called joint spectral radius, as explained in
                  Chapter 4.\par The joint spectral radius analysis proves to
                  be useful in many different contexts like, for example, in
                  the construction of wavelets of compact support, in
                  analysing the asymptotic behavior of solutions of linear
                  difference equations with variable coefficients, in the
                  coordination of autonomous agents and many others. The same
                  quantity, however, can prove to be hard to compute and can
                  lead even to undecidable problems. In this thesis we
                  present all the known generalizations of spectral radius,
                  the properties, theoretical results and challenges
                  associated with them and an algorithm for the exact
                  evaluation of the joint spectral radius. We make use of
                  this algorithm to prove a finiteness conjecture about
                  $2\times 2$ sign-matrices proposed recently by Blondel,
                  Jungers and Protasov.",
}

@MISC{Cicone:ArXiv15,
 author        = "Cicone, Antonio",
 title         = "A note on the Joint Spectral Radius",
 howpublished  = "ArXiv.org e-Print archive",
 year          = "2015",
 month         = feb,
 eprinttype    = "arXiv",
 eprint        = "1502.01506",
 doi           = "10.48550/arXiv.1502.01506",
 annote        = "Last two decades have been characterized by an increasing
                  interest in the analysis of the maximal growth rate of long
                  products generated by matrices belonging to a specific
                  set/family. The maximal growth rate can be evaluated
                  considering a generalization of the spectral radius of a
                  single matrix to the case of a set of matrices.\par This
                  generalization can be formulated in many different ways,
                  nevertheless in the commonly studied cases of bounded or
                  finite families all the possible generalizations coincide
                  in a unique value that is usually called \emph{joint
                  spectral radius} or simply \emph{spectral radius}. The
                  joint spectral radius, however, can prove to be hard to
                  compute and can lead even to undecidable problems. We
                  present in this paper all the possible generalizations of
                  the spectral radius, their properties and the associated
                  theoretical challenges.\par From an historical point of
                  view the first two generalizations of spectral radius, the
                  so--called joint and common spectral radius, were
                  introduced by Rota and Strang in the three pages paper
                  ``\emph{A note on the joint spectral radius}'' published in
                  $1960$~\cite{RotaStr:IM60}. After that more than thirty
                  years had to pass before a second paper was issued on this
                  topic: in $1992$ \emph{I.~Daubechies} and
                  \emph{J.\,C.~Lagarias}~\cite{DaubLag:LAA92} published
                  ``\emph{Sets of matrices all infinite products of which
                  converge}'' introducing the generalized spectral radius,
                  conjecturing it was equal to the joint spectral radius
                  (this was proven immediately after by \emph{M.\,A.~Berger}
                  and \emph{Y.~Wang}~\cite{BerWang:LAA92}) and presenting
                  examples of applications. From then on there has been a
                  rapidly increasing interest on this subject and the more
                  years pass the more the number of mathematical branches and
                  applications directly involved in the study of these
                  quantities increases~\cite{Blondel:LAA08}.\par The study of
                  infinite products convergence properties proves to be of
                  primary interest in a variety of contexts: nonhomogeneous
                  Markov chains, deterministic construction of functions and
                  curves with self-similarities under changes in scale like
                  the von Koch snowflake and the de Rham curves, two-scale
                  refinement equations that arise in the construction of
                  wavelets of compact support and in the dyadic interpolation
                  schemes of Deslauriers and Dubuc, the asymptotic behavior
                  of the solutions of linear difference equations with
                  variable coefficients, coordination of autonomous agents,
                  hybrid systems with applications that range from
                  intelligent traffic systems to industrial process control,
                  the stability analysis of dynamical systems of autonomous
                  differential equations, computer--aided geometric design in
                  constructing parameterized curves and surfaces by
                  subdivision or refinement algorithms, the stability of
                  asynchronous processes in control theory, the stability of
                  desynchronised systems, the analysis of magnetic recording
                  systems and in particular the study of the capacity of
                  codes submitted to forbidden differences constraints,
                  probabilistic automata, the distribution of random power
                  series and the asymptotic behavior of the Euler partition
                  function, the logarithm of the joint spectral radius
                  appears also in the context of discrete linear inclusions
                  as the Lyapunov indicator. For a more extensive and
                  detailed list of applications we refer the reader to the
                  Gilbert Strang's paper ``\emph{The Joint Spectral
                  Radius}''~\cite{Strang03} and to the doctoral theses by
                  \emph{R.~Jungers}~\cite{Jungers:09} and
                  \emph{J.~Theys}~\cite{Theys:PhD05}.\par The paper develops
                  as following: in Section~1 we give notation and terminology
                  used throughout this paper; Section~2 presents first a case
                  of study associated with the asymptotic behavior analysis
                  of the solutions of linear difference equations with
                  variable coefficients, further, it contains the definitions
                  and properties of all the possible generalizations of
                  spectral radius for a set of matrices, in particular the
                  irreducibility, nondefectivity and finiteness properties
                  are discussed.",
}

@ARTICLE{CGP:NAHS18,
 author        = "Cicone, Antonio and Guglielmi, Nicola and Protasov, Vladimir
                  {\relax{}Yu}.",
 authauthor    = "Cicone, Antonio and Guglielmi, Nicola and Protasov,
                  Vladimir",
 title         = "Linear switched dynamical systems on graphs",
 journal       = "Nonlinear Anal. Hybrid Syst.",
 fjournal      = "Nonlinear Analysis. Hybrid Systems",
 year          = "2018",
 volume        = "29",
 pages         = "165--186",
 eprinttype    = "arXiv",
 eprint        = "1607.00415",
 issn          = "1751-570X",
 mrclass       = "93C55 (15A18 37E25 68Q45 93C30 93D20)",
 mrnumber      = "3795595",
 mrreviewer    = "Filippo Cacace",
 zblnumber     = "1391.37035",
 doi           = "10.1016/j.nahs.2018.01.006",
 url           = "https://www.sciencedirect.com/science/article/pii/S1751570X18300116",
 annote        = "We consider linear dynamical systems with a structure of a
                  multigraph. The vertices are associated to linear spaces
                  and the edges correspond to linear maps between those
                  spaces. We analyse the asymptotic growth of trajectories
                  (associated to paths along the multigraph), the stability
                  and the stabilizability problems. This generalizes the
                  classical linear switching systems and their recent
                  extensions to Markovian systems, to systems generated by
                  regular languages, etc. We show that an arbitrary system
                  can be factorized into several irreducible systems on
                  strongly connected multigraphs. For the latter systems, we
                  prove the existence of invariant (Barabanov) multinorm and
                  derive a method of its construction. The method works for a
                  vast majority of systems and finds the joint spectral
                  radius (Lyapunov exponent). Numerical examples are
                  presented and applications to the study of fractals,
                  attractors, and multistep methods for ODEs are discussed.",
}

@ARTICLE{CGSZ:LAA10,
 author        = "Cicone, Antonio and Guglielmi, Nicola and Serra-Capizzano,
                  Stefano and Zennaro, Marino",
 title         = "Finiteness property of pairs of {$2\times 2$} sign-matrices
                  via real extremal polytope norms",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2010",
 volume        = "432",
 number        = "2-3",
 pages         = "796--816",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15B35 (15A18 15A60)",
 mrnumber      = "2577718 (2011a:15047)",
 mrreviewer    = "Rajesh Pereira",
 zblnumber     = "1186.15006",
 zblreviewer   = "Petko Hr. Petkov (Sofia)",
 doi           = "10.1016/j.laa.2009.09.022",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379509004972",
 annote        = "This paper deals with the joint spectral radius of a finite
                  set of matrices. We say that a set of matrices has the
                  finiteness property if the maximal rate of growth, in the
                  multiplicative semigroup it generates, is given by the
                  powers of a finite product.\par Here we address the problem
                  of establishing the finiteness property of pairs of
                  $n\times n$ sign-matrices. Such problem is related to the
                  conjecture that pairs of sign-matrices fulfil the
                  finiteness property for any dimension. This would imply, by
                  a recent result by Blondel and Jungers, that finite sets of
                  rational matrices fulfil the finiteness property, which
                  would be very important in terms of the computation of the
                  joint spectral radius. The technique used in this paper
                  could suggest an extension of the analysis to $n\times n$
                  sign-matrices, which still remains an open problem.\par As
                  a main tool of our proof we make use of a procedure to find
                  a so-called real extremal polytope norm for the set. In
                  particular, we present an algorithm which, under some
                  suitable assumptions, is able to check if a certain product
                  in the multiplicative semigroup is spectrum maximizing.\par
                  For pairs of sign-matrices we develop the computations
                  exactly and hence are able to prove analytically the
                  finiteness property. On the other hand, the algorithm can
                  be used in a floating point arithmetic and provide a
                  general tool for approximating the joint spectral radius of
                  a set of matrices.",
}

@MISC{JSRpack,
 author        = "Cicone, Antonio and Protasov, Vladimir",
 title         = "Joint spectral radius computation",
 howpublished  = "{MATLAB}{$^\circledR$} Central",
 year          = "2012",
 url           = "https://www.mathworks.com/matlabcentral/fileexchange/36460-joint-spectral-radius-computation",
 annote        = "The algorithm allows to evaluate lower and upper bounds for
                  the JSR of a set of matrices and estimates its candidate
                  spectrum maximizing products (s.m.p). It is based on
                  ellipsoidal norms and a branch and bound technique.",
}

@ARTICLE{Cohen:MPCPS79,
 author        = "Cohen, Joel E.",
 authauthor    = "Cohen, Joel",
 title         = "Contractive inhomogeneous products of nonnegative matrices",
 journal       = "Math. Proc. Camb. Philos. Soc.",
 fjournal      = "Mathematical Proceedings of the Cambridge Philosophical
                  Society",
 year          = "1979",
 volume        = "86",
 issue         = "02",
 pages         = "351--364",
 pagetotal     = "14",
 month         = sep,
 issn          = "1469-8064",
 mrclass       = "15A48 (60J10)",
 mrnumber      = "538757",
 mrreviewer    = "Author's review",
 zblnumber     = "0415.60013",
 doi           = "10.1017/S0305004100056176",
 url           = "https://dx.doi.org/10.1017/S0305004100056176",
 annote        = "\emph{J.~Hajnal}~\cite{Hajnal:MPCPS76} showed that under
                  wide conditions a sequence of products $H(1, q) =
                  M_{q}\cdots M_{1}$, $q = 1,2,\ldots $, of square
                  non-negative matrices $M_{q}$ approaches a sequence of
                  positive matrices of rank $1$. We call a product $H(1,q)$
                  inhomogeneous if its factors $M_{1},\ldots, M_{q}$ are not
                  necessarily all equal to one another. When the matrices
                  $M_{q}$ are members of an `ergodic set', and $x$ and $y$
                  are positive vectors, the projective distance $d(H(1,q)x,
                  H(1,q)y)$ decays at least exponentially fast as $q$
                  increases. An important condition on an ergodic set is that
                  any product of $g$ members from the set be a matrix in
                  which all elements are (strictly) positive, where $g$ is
                  some fixed positive integer.",
}

@ARTICLE{Cohn:IJMMS,
 author        = "Cohn, Harry",
 title         = "Products of stochastic matrices and applications",
 journal       = "Internat. J. Math. Math. Sci.",
 fjournal      = "International Journal of Mathematics and Mathematical
                  Sciences",
 year          = "1989",
 volume        = "12",
 number        = "2",
 pages         = "209--233",
 issn          = "0161-1712",
 mrclass       = "15A51 (60J10)",
 mrnumber      = "994904 (90e:15033)",
 mrreviewer    = "Ray C. Shiflett",
 zblnumber     = "0673.15010",
 zblreviewer   = "Eugene Seneta (Sydney)",
 doi           = "10.1155/S0161171289000268",
 url           = "https://www.hindawi.com/journals/ijmms/1989/656040/",
 annote        = "The paper deals with aspects of the limit behaviour of
                  products of nonidentical finite or countable stochastic
                  matrices $(P_n)$. Applications are given to nonhomogeneous
                  Markov models as positive chains, some classes of finite
                  chains considered by Doeblin and weakly ergodic chains.\par
                  This paper is a streamlined survey of the literature of
                  non-homogeneous Markov chains from the standpoint of tail
                  $\sigma$-fields. The original papers were in the main the
                  author's own [Math. Proc. Camb. Phil. Soc. 92, 527--534
                  (1982); Adv. Appl. Prob. 8, 502--516 (1976) and 9, 542--552
                  (1977) and 13, 388--401 (1981)].",
}

@ARTICLE{Cong:SCL16,
 author        = "Cong, Shen",
 title         = "Stability analysis for planar discrete-time linear switching
                  systems via bounding joint spectral radius",
 journal       = "Systems Control Lett.",
 fjournal      = "Systems {\&} Control Letters",
 year          = "2016",
 volume        = "96",
 pages         = "7--10",
 issn          = "0167-6911",
 mrclass       = "93D09 (15A60 39A60 93C55)",
 mrnumber      = "3547649",
 mrreviewer    = "Jacek Kabzi{\'n}ski",
 zblnumber     = "1347.93230",
 doi           = "10.1016/j.sysconle.2016.06.012",
 url           = "https://www.sciencedirect.com/science/article/pii/S0167691116300780",
 annote        = "A pair of 2-by-2 matrices can be transformed into the joint
                  forms, which are bound together via an invariant that
                  measures the deformation of their structures with respect
                  to each other. With this, we obtain an approximation of the
                  joint spectral radius of such a pair of matrices from above
                  and then apply it to assessing stability for planar
                  discrete-time linear switching systems.",
}

@ARTICLE{CrossKPP:MECA12,
 author        = "Cross, Rod and Kozyakin, Victor and O'Callaghan, Brian and
                  Pokrovskii, Alexei and Pokrovskiy, Alexey",
 title         = "Periodic Sequences of Arbitrage: A Tale of Four Currencies",
 journal       = "Metroeconomica",
 year          = "2012",
 volume        = "63",
 number        = "2",
 pages         = "250--294",
 month         = may,
 eprinttype    = "arXiv",
 eprint        = "1112.5850",
 zblnumber     = "1242.91152",
 doi           = "10.1111/j.1467-999X.2011.04140.x",
 url           = "https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1467-999X.2011.04140.x",
 annote        = "This paper investigates arbitrage chains involving four
                  currencies and four foreign exchange trader-arbitrageurs.
                  In contrast with the three-currency case, we find that
                  arbitrage operations when four currencies are present may
                  appear periodic in nature, and not involve smooth
                  convergence to a `balanced' ensemble of exchange rates in
                  which the law of one price holds. The goal of this article
                  is to understand some interesting features of sequences of
                  arbitrage operations, features of which might well be
                  relevant in other contexts in finance and economics.",
}

@ARTICLE{CrossKoz:DCDSB13,
 author        = "Cross, Rod and Kozyakin, Victor S.",
 authauthor    = "Cross, Rod and Kozyakin, Victor",
 title         = "Double exponential instability of triangular arbitrage
                  systems",
 journal       = "Discrete Contin. Dyn. Syst. Ser. B",
 fjournal      = "Discrete and Continuous Dynamical Systems. Series B. A
                  Journal Bridging Mathematics and Sciences",
 year          = "2013",
 volume        = "18",
 number        = "2",
 pages         = "349--376",
 eprinttype    = "arXiv",
 eprint        = "1204.3422",
 issn          = "1531-3492",
 mrclass       = "91G80",
 mrnumber      = "2999081",
 zblnumber     = "1260.91261",
 doi           = "10.3934/dcdsb.2013.18.349",
 url           = "https://www.aimsciences.org/article/doi/10.3934/dcdsb.2013.18.3499",
 annote        = "If financial markets displayed the informational efficiency
                  postulated in the efficient markets hypothesis (EMH),
                  arbitrage operations would be self-extinguishing. The
                  present paper considers arbitrage sequences in foreign
                  exchange (FX) markets, in which trading platforms and
                  information are fragmented. In~\cite{CrossKPP:MECA12} it
                  was shown that sequences of triangular arbitrage operations
                  in FX markets containing 4 currencies and
                  trader-arbitrageurs tend to display periodicity or grow
                  exponentially rather than being self-extinguishing. This
                  paper extends the analysis to 5 or higher-order currency
                  worlds. The key findings are that in a 5-currency world
                  arbitrage sequences may also follow an exponential law as
                  well as display periodicity, but that in higher-order
                  currency worlds a double exponential law may additionally
                  apply. There is an ``inheritance of instability'' in the
                  higher-order currency worlds. Profitable arbitrage
                  operations are thus endemic rather that displaying the
                  self-extinguishing properties implied by the EMH.",
}

@ARTICLE{ProtCvet:MP20,
 author        = "Cvetkovi{\'c}, Aleksandar and Protasov, Vladimir
                  {\relax{}Yu}.",
 authauthor    = "Cvetkovi{\'c}, Aleksandar and Protasov, Vladimir",
 title         = "The greedy strategy in optimizing the {P}erron eigenvalue",
 journal       = "Math. Program.",
 year          = "2022",
 volume        = "193",
 number        = "1, Ser. A",
 pages         = "1--31",
 eprinttype    = "arXiv",
 eprint        = "1807.05099",
 issn          = "0025-5610",
 mrclass       = "15B48 (15A42 90C26 90C40)",
 mrnumber      = "4412387",
 zblnumber     = "07516304",
 doi           = "10.1007/s10107-020-01585-z",
 url           = "https://link.springer.com/article/10.1007/s10107-020-01585-z",
 annote        = "We address the problems of minimizing and of maximizing the
                  spectral radius over a convex family of non-negative
                  matrices. Those problems being hard in general can be
                  efficiently solved for some special families. We consider
                  the so-called product families, where each matrix is
                  composed of rows chosen independently from given sets. A
                  recently introduced greedy method works surprisingly fast.
                  However, it is applicable mostly for strictly positive
                  matrices. For sparse matrices, it often diverges and gives
                  a wrong answer. We present the ``selective greedy method''
                  that works equally well for all non-negative product
                  families, including sparse ones. For this method, we prove
                  a quadratic rate of convergence and demonstrate its
                  exceptional efficiency in numerical examples. In dimensions
                  up to 2000, the matrices with minimal/maximal spectral
                  radii in product families are found within a few
                  iterations. Applications to dynamical systems and to the
                  graph theory are considered.",
}

@INPROCEEDINGS{CzorNaw:CCA07,
 author        = "Czornik, A. and Nawrat, A.",
 title         = "Generalized spectral radius and {L}yapunov exponents of
                  linear time varying systems",
 booktitle     = "IEEE International Conference on Control Applications, 2007.
                  CCA 2007",
 publisher     = "IEEE",
 address       = "Singapore",
 year          = "2007",
 pages         = "1303--1306",
 doi           = "10.1109/CCA.2007.4389415",
 url           = "https://ieeexplore.ieee.org/document/4389415",
 annote        = "In this paper we consider discrete time varying linear
                  systems with matrix coefficients in fixed set and we
                  describe the set of all possible Lyapunov exponents for the
                  system. We describe this set in terms of generalized
                  spectral sub radius and the generalized spectral radius of
                  the set of possible values of the coefficients.",
}

@ARTICLE{Czornik:LAA05,
 author        = "Czornik, Adam",
 title         = "On the generalized spectral subradius",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2005",
 volume        = "407",
 pages         = "242--248",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15A18",
 mrnumber      = "2161929 (2006d:15021)",
 zblnumber     = "1080.15008",
 zblreviewer   = "C. M. da Fonseca (Coimbra)",
 doi           = "10.1016/j.laa.2005.05.006",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379505002582",
 annote        = "For $m\ge 1$, let $\Sigma^{m}$ be the set of all products of
                  length $m$ of square complex matrices of some order $k$.
                  Let $\rho(A)$ and $\|A\|$ be, respectively, the spectral
                  radius and a matrix norm of a matrix $A$. In this note the
                  author proves that the joint and the generalized spectral
                  subradius are equal to infimum limits of
                  $\inf_{A\in\Sigma^{m}} \|A\|^{1/m}$ and
                  $\inf_{A\in\Sigma^{m}}\rho^{1/m}(A)$. This generalizes to
                  the case of infinite set of matrices a formula due to
                  \emph{L.~Gurvits}~\cite{Gurv:LAA95}. A new definition of
                  stability of discrete linear inclusions is proposed.",
}

@ARTICLE{CJ:IJAMCS06,
 author        = "Czornik, Adam and Jurga{\'s}, Piotr",
 title         = "Some properties of the spectral radius of a set of
                  matrices",
 journal       = "Int. J. Appl. Math. Comput. Sci.",
 fjournal      = "International Journal of Applied Mathematics and Computer
                  Science",
 year          = "2006",
 volume        = "16",
 number        = "2",
 pages         = "183--188",
 issn          = "1641-876X",
 mrclass       = "15A18 (15A42 93D09)",
 mrnumber      = "2238011 (2007d:15016)",
 mrreviewer    = "Vakif Y. Dzhafarov",
 zblnumber     = "1113.15009",
 zblreviewer   = "Grosio Stanilov (Sofia)",
 url           = "http://matwbn.icm.edu.pl/ksiazki/amc/amc16/amc1623.pdf",
 annote        = "We show new formulas for the spectral radius and the
                  spectral subradius of a set of matrices. The advantage of
                  our results is that we express the spectral radius of any
                  set of matrices by the spectral radius of a set of
                  symmetric positive definite matrices. In particular, in one
                  of our formulas the spectral radius is expressed by
                  singular eigenvalues of matrices, whereas in the existing
                  results it is expressed by eigenvalues.",
}

@ARTICLE{CJ:IJAMCS07,
 author        = "Czornik, Adam and Jurga{\'s}, Piotr",
 title         = "Falseness of the finiteness property of the spectral
                  subradius",
 journal       = "Int. J. Appl. Math. Comput. Sci.",
 fjournal      = "International Journal of Applied Mathematics and Computer
                  Science",
 year          = "2007",
 volume        = "17",
 number        = "2",
 pages         = "173--178",
 issn          = "1641-876X",
 mrclass       = "15A18 (93C55 93D20)",
 mrnumber      = "2341291 (2008f:15035)",
 zblnumber     = "1126.93028",
 zblreviewer   = "Mihail Voicu (Ia{\c{s}}i)",
 doi           = "10.2478/v10006-007-0016-1",
 url           = "https://content.sciendo.com/view/journals/amcs/17/2/article-p173.xml",
 annote        = "By using only elementary facts from the theory of formal
                  languages and from linear algebra, it is proved (but not in
                  a constructive manner) that there exist infinitely many
                  values of the real parameter $\alpha$ for which the exact
                  value of the spectral subradius of the pair of matrices
                  $\left[\begin{smallmatrix} 1 & 1 \\ 0 & 1 \\
                  \end{smallmatrix}\right]$, $\alpha
                  \left[\begin{smallmatrix} 1 & 0 \\ 1 & 1 \\
                  \end{smallmatrix}\right]$ cannot be calculated in a finite
                  number of steps. This kind of falseness does not imply that
                  no algorithm exists for an exact value computing of the
                  spectral subradius of a finite set of real matrices in a
                  finite number of steps. It simply states that it is
                  impossible to set forth such an algorithm in the way that
                  is suggested in the finiteness property of the spectral
                  subradius. Thus, the problem of inventing such an algorithm
                  is still open.",
}

@ARTICLE{CzorJurg:Automatica08,
 author        = "Czornik, Adam and Jurga{\'s}, Piotr",
 title         = "Set of possible values of maximal {L}yapunov exponents of
                  discrete time-varying linear system",
 journal       = "Automatica J. IFAC",
 fjournal      = "Automatica. A Journal of IFAC, the International Federation
                  of Automatic Control",
 year          = "2008",
 volume        = "44",
 number        = "2",
 pages         = "580--583",
 issn          = "0005-1098",
 mrclass       = "93C55 (34D08 39A30)",
 mrnumber      = "2530809",
 zblnumber     = "1283.93170",
 doi           = "10.1016/j.automatica.2007.06.028",
 url           = "https://www.sciencedirect.com/science/article/pii/S0005109807003433?via\%3Dihub",
 annote        = "In this paper we consider discrete time varying linear
                  systems with coefficients in fixed set of invertible
                  matrices and we describe the set of all possible maximal
                  Lyapunov exponents for the system. We show that the set
                  includes an interval bounded by the generalized spectral
                  subradius and the generalized spectral radius.",
}

@INCOLLECTION{CJN:AISC15,
 author        = "Czornik, Adam and Jurga{\'s}, Piotr and Niezabitowski,
                  Micha{\l}",
 editor        = "Gruca, Aleksandra and Brachman, Agnieszka and Kozielski,
                  Stanis{\l}aw and Czach{\'o}rski, Tadeusz",
 title         = "Estimation of the Joint Spectral Radius",
 booktitle     = "Man-Machine Interactions 4",
 series        = "Advances in Intelligent Systems and Computing",
 publisher     = "Springer International Publishing",
 year          = "2016",
 volume        = "391",
 pages         = "401--410",
 isbn          = "978-3-319-23436-6",
 doi           = "10.1007/978-3-319-23437-3_34",
 url           = "https://link.springer.com/chapter/10.1007/978-3-319-23437-3_34",
 language      = "English",
 annote        = "The joint spectral radius of a set of matrices is a
                  generalization of the concept of spectral radius of a
                  matrix. Such notation has many applications in the computer
                  science, and more generally in applied mathematics. It has
                  been used, for example in graph theory, control theory,
                  capacity of codes, continuity of wavelets, overlap-free
                  words, trackable graphs. It is impossible to provide
                  analytic formulae for this quantity and therefore any
                  estimation are highly desired. The main result of this
                  paper is to provide an estimation of the joint spectral
                  radius in the terms of matrices norms and spectral radii.",
}

@INPROCEEDINGS{CzorNiez:MMAR13,
 author        = "Czornik, Adam and Niezabitowski, Michal{\l}",
 title         = "Controllability and stability of switched systems",
 booktitle     = "18th International Conference on Methods and Models in
                  Automation and Robotics (MMAR)",
 publisher     = "IEEE",
 address       = "26-29 Aug. 2013, Miedzyzdroje, Poland",
 year          = "2013",
 pages         = "16--21",
 doi           = "10.1109/MMAR.2013.6669874",
 url           = "https://ieeexplore.ieee.org/document/6669874",
 annote        = "The study of properties of switched and hybrid systems gives
                  rise to a number of interesting and challenging
                  mathematical problems. This paper aim to briefly survey
                  recent results on stability and controllability of switched
                  linear systems. First, the stability analysis for switched
                  systems is reviewed. We focus on the stability analysis for
                  switched linear systems under arbitrary switching, and we
                  highlight necessary and sufficient conditions for
                  asymptotic stability.",
}

@ARTICLE{CzorNiez:JFI15,
 author        = "Czornik, Adam and Niezabitowski, Michal{\l}",
 title         = "Alternative formulae for lower general exponent of discrete
                  linear time-varying systems",
 journal       = "J. Franklin Inst.",
 fjournal      = "Journal of the Franklin Institute. Engineering and Applied
                  Mathematics",
 year          = "2015",
 volume        = "352",
 number        = "1",
 pages         = "399--419",
 issn          = "0016-0032",
 mrclass       = "37N35 (93C55)",
 mrnumber      = "3292336",
 zblnumber     = "1307.93237",
 doi           = "10.1016/j.jfranklin.2014.11.003",
 url           = "https://www.sciencedirect.com/science/article/pii/S0016003214003196",
 annote        = "The Bohl exponents, similarly as Lyapunov exponents, are one
                  of the most important numerical characteristics of
                  dynamical systems used in control theory. Properties of the
                  Lyapunov characteristics are well described in the
                  literature. Properties of the Bohl exponents are much less
                  investigated. In this paper we consider the so-called
                  junior lower general exponent of discrete linear
                  time-varying system and present some alternative formulae
                  for it. We also discuss relations between lower Bohl
                  exponents of the perturbed system and junior lower general
                  exponent of the unperturbed system.",
}

@MISC{Dai:ArXiv11,
 author        = "Dai, Xiongping",
 title         = "A criterion of simultaneously symmetrization and spectral
                  finiteness for a finite set of real 2-by-2 matrices",
 howpublished  = "ArXiv.org e-Print archive",
 year          = "2011",
 month         = nov,
 eprinttype    = "arXiv",
 eprint        = "1111.2108",
 doi           = "10.48550/arXiv.1111.2108",
 annote        = "In this paper, we consider the simultaneously symmetrization
                  and spectral finiteness for a finite set of real 2-by-2
                  matrices.",
}

@ARTICLE{Dai:JMAA11,
 author        = "Dai, Xiongping",
 title         = "Extremal and {B}arabanov semi-norms of a semigroup generated
                  by a bounded family of matrices",
 journal       = "J. Math. Anal. Appl.",
 fjournal      = "Journal of Mathematical Analysis and Applications",
 year          = "2011",
 volume        = "379",
 number        = "2",
 pages         = "827--833",
 coden         = "JMANAK",
 issn          = "0022-247X",
 mrclass       = "15A60",
 mrnumber      = "2784362 (2012d:15045)",
 mrreviewer    = "Sing Cheong Ong",
 zblnumber     = "1215.15025",
 doi           = "10.1016/j.jmaa.2010.12.059",
 url           = "https://www.sciencedirect.com/science/article/pii/S0022247X11000126",
 annote        = "Let $S=\{S_i\}_{i \in \mathcal{I}}$ be, an, arbitrary family
                  of complex $n$-by-$n$ matrices, where $1 \leq n < \infty$.
                  Let $\hat{\rho}(S)$ denote the joint spectral radius of
                  $S$, defined as \[\hat{\rho}(S)=\limsup\left\{
                  \sup_{(i_1,\ldots , i_l)\in \mathcal{I}^l} \| S_{i_1}\cdots
                  S_{i_l}\|^{1/l}\right\},\] which is independent of the norm
                  $\|\cdot\|$ used here. A semi-norm $\|\cdot\|_{*}$ on
                  $\mathbb{C}^n$ is called ``extremal'' of $S$, if it
                  satisfies \[\|x\|_{*}\not\equiv 0\qquad\text{and}\qquad
                  \|x\cdot S_i\|_{*} \le\hat{\rho}(S)\|x\|_{*}\quad\forall x
                  = (x_1,\ldots,x_n)\in\mathbb{C}^n~\text{and}~ i\in
                  \mathcal{I}.\] In this paper, using an elementary analytic
                  approach the author shows that if $S$ is bounded in
                  $\mathbb{C}^{n\times n}$, then there always exists, for
                  $S$, an extremal semi-norm $\|\cdot\|_{*}$ on
                  $\mathbb{C}^n$; if additionally $S$ is compact in
                  $(\mathbb{C}^{n\times n}$, $\| \, \cdot \, \|)$, this
                  extremal semi-norm has the ``Barabanov-type property'',
                  i.e., to any $x \in \mathbb{C}^n$, one can find an infinite
                  sequence $i_{\mathbf{\cdot}}: \mathbb{N}\rightarrow
                  \mathcal{I}$ with $\| x \cdot S_{i_1}\cdots
                  S_{i_k}\|_*=\hat{\rho}(S)^k\| x \|_{*}$ for each $k\geq 1$.
                  This implies and generalizes the Barabanov's Norm Theorem,
                  Berger--Wang's Formula and Elsner's Reduction Theorem.",
}

@ARTICLE{Dai:N11,
 author        = "Dai, Xiongping",
 title         = "Optimal state points of the subadditive ergodic theorem",
 journal       = "Nonlinearity",
 fjournal      = "Nonlinearity",
 year          = "2011",
 volume        = "24",
 number        = "5",
 pages         = "1565--1573",
 coden         = "NONLE5",
 issn          = "0951-7715",
 mrclass       = "37A30 (28D05 37H15)",
 mrnumber      = "2785982 (2012f:37008)",
 mrreviewer    = "Ian Melbourne",
 zblnumber     = "1254.37007",
 doi           = "10.1088/0951-7715/24/5/009",
 url           = "https://iopscience.iop.org/article/10.1088/0951-7715/24/5/009",
 annote        = "Let $\boldsymbol{S}=\{S_1,\dotsc,S_K\}$ be a finite set of
                  $d\times d$ complex matrices with the joint spectral radius
                  1. In this paper, as a corollary of an ergodic theorem
                  proved there, it is proved the fillowing:\par
                  \textbf{Corollary 4.} \emph{If $\boldsymbol{S}$ is
                  periodically switched stable and $\sigma=(i_1,i_2,\dotsc)$
                  in $\{1,\dotsc,K\}^\mathbb{N}$ is a switching law which
                  generates a Markovian measure under the standard shift
                  transformation, then $S_{i_1}\dotsm S_{i_n}\to 0$ as
                  $n\to\infty$.}",
}

@ARTICLE{Dai:JDE11,
 author        = "Dai, Xiongping",
 title         = "Weakly {B}irkhoff recurrent switching signals, almost sure
                  and partial stability of linear switched dynamical
                  systems",
 journal       = "J. Differential Equations",
 fjournal      = "Journal of Differential Equations",
 year          = "2011",
 volume        = "250",
 number        = "9",
 pages         = "3584--3629",
 coden         = "JDEQAK",
 issn          = "0022-0396",
 mrclass       = "93D05 (37N35 93C15 93C30)",
 mrnumber      = "2773179 (2012e:93109)",
 mrreviewer    = "Mahesh Nerurkar",
 zblnumber     = "1208.93045",
 doi           = "10.1016/j.jde.2011.01.029",
 url           = "https://www.sciencedirect.com/science/article/pii/S0022039611000520",
 annote        = "Let $\mathcal{I}$ be a separable metric space, not
                  necessarily compact, and
                  $\boldsymbol{S}\colon\mathcal{I}\rightarrow\mathbb{C}^{d\times
                  d}$ be a continuous bounded function with the joint
                  spectral radius 1. We denote by
                  $\varSigma_\mathcal{I}^+=\{\sigma_{\cdot}\colon\mathbb{N}\rightarrow\mathcal{I}\}$
                  the product space $\mathcal{I}^\mathbb{N}$ and let
                  $\theta_+\colon\sigma_{\cdot}\mapsto\sigma_{\cdot+1}$ be
                  the standard shift transformation. One of the main theorems
                  is the following:\par \textbf{Theorem C'.} \emph{If $\mu$
                  is an ergodic measure of $\theta_+$ and the support of
                  $\mu$ contains a stable switching law $\sigma_{\cdot}$ for
                  $\boldsymbol{S}$, then $\boldsymbol{S}$ is $\mu$-almost
                  surely exponentially stable.}",
}

@ARTICLE{Dai:LAA12,
 author        = "Dai, Xiongping",
 title         = "A {G}el'fand-type spectral-radius formula and stability of
                  linear constrained switching systems",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2012",
 volume        = "436",
 number        = "5",
 pages         = "1099--1113",
 eprinttype    = "arXiv",
 eprint        = "1107.0124",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15A60 (93D20)",
 mrnumber      = "2890907",
 mrreviewer    = "Yun Zou",
 zblnumber     = "1237.15010",
 zblreviewer   = "C. M. da Fonseca (Coimbra)",
 doi           = "10.1016/j.laa.2011.07.029",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379511005416",
 annote        = "Using ergodic theory, in this paper we present a
                  Gel'fand-type spectral radius formula which states that the
                  joint spectral radius is equal to the generalized spectral
                  radius for a matrix multiplicative semigroup
                  $\boldsymbol{S}^+$ restricted to a subset that need not
                  carry the algebraic structure of $\boldsymbol{S}^+$. This
                  generalizes the Berger--Wang formula. Using it as a tool,
                  we study the absolute exponential stability of a linear
                  switched system driven by a compact subshift of the
                  one-sided Markov shift associated to $\boldsymbol{S}$ (cf.
                  \emph{I.\,D.~Morris}~\cite{Morris:JFA12}).",
}

@ARTICLE{Dai:LAA13,
 author        = "Dai, Xiongping",
 title         = "Some criteria for spectral finiteness of a finite subset of
                  the real matrix space {$\mathbb{R}^{d\times d}$}",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2013",
 volume        = "438",
 number        = "6",
 pages         = "2717--2727",
 eprinttype    = "arXiv",
 eprint        = "1206.2110",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15A18 (65F15 93D20)",
 mrnumber      = "3008528",
 mrreviewer    = "Hiroshi Nakazato",
 zblnumber     = "1270.15006",
 doi           = "10.1016/j.laa.2012.09.026",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379512007409",
 annote        = "In this paper, we present some checkable criteria for the
                  spectral finiteness of a finite subset of the real $d\times
                  d$ matrix space $\mathbb{R}^{d\times d}$, where $2\le
                  d<\infty$.",
}

@ARTICLE{Dai:JFI14,
 author        = "Dai, Xiongping",
 title         = "Robust periodic stability implies uniform exponential
                  stability of {M}arkovian jump linear systems and random
                  linear ordinary differential equations",
 journal       = "J. Franklin Inst.",
 fjournal      = "Journal of the Franklin Institute. Engineering and Applied
                  Mathematics",
 year          = "2014",
 volume        = "351",
 number        = "5",
 pages         = "2910--2937",
 month         = may,
 eprinttype    = "arXiv",
 eprint        = "1307.4209",
 issn          = "0016-0032",
 mrclass       = "93E15 (34D20 37H10 60J10)",
 mrnumber      = "3191925",
 zblnumber     = "1372.93209",
 doi           = "10.1016/j.jfranklin.2014.01.010",
 url           = "https://www.sciencedirect.com/science/article/pii/S001600321400012X",
 annote        = "In this paper, there are shown the following two statements.
                  \begin{enumerate} \item[(1)] \emph{A discrete-time
                  Markovian jump linear system is uniformly exponentially
                  stable if and only if it is robustly periodically stable,
                  by using a Gel'fand--Berger--Wang formula proved here.}
                  \item[(2)] \emph{A random linear ODE driven by a semiflow
                  with closing by periodic orbits property is uniformly
                  exponentially stable if and only if it is robustly
                  periodically stable, by using Shantao Liao's perturbation
                  technique and the semi-uniform ergodic theorems.}
                  \end{enumerate} The proofs involve ergodic theory in both
                  of the above two cases. In addition, counterexamples are
                  constructed to the robustness condition and to spectral
                  finiteness of linear cocycle.",
}

@MISC{DHH:ArXiv14,
 author        = "Dai, Xiongping and Huang, Tingwen and Huang, Yu",
 title         = "Exponential stability of nonhomogeneous matrix-valued
                  {M}arkovian chains",
 howpublished  = "ArXiv.org e-Print archive",
 year          = "2014",
 month         = jan,
 eprinttype    = "arXiv",
 eprint        = "1401.6014",
 doi           = "10.48550/arXiv.1401.6014",
 annote        = "Let $\boldsymbol{\xi}=\{\xi_n\}_{n\ge0}$ be a
                  nonhomogeneous/nonstationary Markovian chain on a
                  probability space $(\Omega,\mathscr{F},\mathbb{P})$ valued
                  in the state space $\boldsymbol{S}$ that consists of a
                  finite number of real $d$-by-$d$ matrices such that
                  $\mathbb{P}(\{\xi_0=S\})>0$ for each $S\in\boldsymbol{S}$.
                  As usual, $\boldsymbol{\xi}$ is called \textit{uniformly
                  exponentially stable} if there exist two constants $C>0$
                  and $0<\lambda<1$ so that for all $n\ge1$,
                  $\|\xi_0(\omega)\xi_1(\omega)\dotsm\xi_{n-1}(\omega)\|\le
                  C\lambda^{n}$ for $\mathbb{P}\textrm{-a.e.}
                  \omega\in\Omega$. In this note, we show that if the
                  Markovian transition probability matrices of
                  $\boldsymbol{\xi}$ have the same transition sign matrix for
                  all times $n\ge0$, then $\boldsymbol{\xi}$ is uniformly
                  exponentially stable if and only if there are $\gamma<1$
                  and $N>0$ such that for each $n>N$, the spectral radii
                  $\rho(S_{i_0}\dotsm S_{i_{n-1}})$ are less than or equal to
                  $\gamma$ for all $n$-length closed sample paths
                  $(S_{i_0},\dotsm, S_{i_{n-1}})\in\boldsymbol{S}^n$ with
                  $\mathbb{P}(\{\xi_0=S_{i_0},\dotsc,\xi_{n-1}=S_{i_{n-1}},
                  \xi_n=S_{i_0}\})>0$.",
}

@ARTICLE{DHHX:JFI17,
 author        = "Dai, Xiongping and Huang, Tingwen and Huang, Yu and Luo, Yi
                  and Wang, Gang and Xiao, Mingqing",
 title         = "Chaotic behavior of discrete-time linear inclusion dynamical
                  systems",
 journal       = "J. Franklin Inst.",
 fjournal      = "Journal of the Franklin Institute. Engineering and Applied
                  Mathematics",
 year          = "2017",
 volume        = "354",
 number        = "10",
 pages         = "4126--4155",
 eprinttype    = "arXiv",
 eprint        = "1308.4274",
 issn          = "0016-0032",
 mrclass       = "37D45",
 mrnumber      = "3651331",
 zblnumber     = "1367.93270",
 doi           = "10.1016/j.jfranklin.2017.03.010",
 url           = "https://www.sciencedirect.com/science/article/abs/pii/S0016003217301473",
 annote        = "Given $K$ real $d$-by-$d$ nonsingular matrices
                  $S_1,\ldots,S_K$, by extending the well-known Li--Yorke
                  chaotic description of a deterministic nonlinear dynamical
                  system to a discrete-time linear inclusion dynamical
                  system: $x_n\in\left\{S_kx_{n-1}\right\}_{1\le k\le K}$
                  with $x_0\in\mathbb{R}^d$ and $n\ge1$, we study the chaotic
                  characteristic of the state trajectory
                  $(x_n(x_0,\sigma))_{n\ge1}$, governed by a switching law
                  $\sigma\colon\mathbb{N}\rightarrow\{1,\ldots,K\}$, for any
                  initial states $x_0\in\mathbb{R}^d$. Two sufficient
                  conditions are given so that for a ``large'' subset of the
                  space of all possible switching laws $\sigma$, we have the
                  sharp infinite oscillation as follows: \begin{equation*}
                  \liminf\limits_{n\to+\infty}\|x_n(x_0,\sigma)\|=0
                  \quad\textrm{and}\quad
                  \limsup\limits_{n\to+\infty}\|x_n(x_0,\sigma)\|=+\infty\quad
                  \forall x_0\in\mathbb{R}^d\setminus\{0\}. \end{equation*}
                  This implies that there coexists at least one positive, one
                  zero and one negative Lyapunov exponents and that the
                  trajectories $(x_n(x_0,\sigma))_{n\ge1}$ are extremely
                  sensitive to the initial states $x_0\in\mathbb{R}^d$. We
                  also show that a periodically stable linear inclusion
                  system, which may be unbounded, does not have any such
                  chaotic behavior.",
}

@ARTICLE{DHLX:LAA12,
 author        = "Dai, Xiongping and Huang, Yu and Liu, Jun and Xiao,
                  Mingqing",
 title         = "The finite-step realizability of the joint spectral radius
                  of a pair of {$d\times d$} matrices one of which being
                  rank-one",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2012",
 volume        = "437",
 number        = "7",
 pages         = "1548--1561",
 eprinttype    = "arXiv",
 eprint        = "1106.0870",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15A60 (93D20)",
 mrnumber      = "2946341",
 mrreviewer    = "Anar Adigezal Dosiev",
 zblnumber     = "1250.15011",
 zblreviewer   = "Jorma K. Merikoski (Tampere)",
 doi           = "10.1016/j.laa.2012.04.053",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379512003564",
 annote        = "We study the finite-step realizability of the,
                  joint/generalized spectral radius of a pair of real
                  $d\times d$ matrices $\{\mathrm{S}_1,\mathrm{S}_2\}$, one
                  of which has rank $1$, where $2\le d<+\infty$. Let
                  $\rho(A)$ denote the spectral radius of a square matrix
                  $A$. Then we prove that there always exists a finite-length
                  word $({i}_1^*,\ldots,{i}_\ell^*)\in\{1,2\}^\ell$, for some
                  finite $\ell\ge1$, such that \begin{equation*}
                  \sqrt[\ell]{\rho(\mathrm{S}_{{i}_1^*}\cdots
                  \mathrm{S}_{{i}_\ell^*})}=
                  \sup_{n\ge1}\left\{\max_{(i_1,\ldots,i_n)\in
                  \{1,2\}^n}\sqrt[n]{\rho(\mathrm{S}_{i_1}\cdots
                  \mathrm{S}_{i_n})}\right\}; \end{equation*} that is to say,
                  there holds the spectral finiteness property for
                  $\{\mathrm{S}_1,\mathrm{S}_2\}$. This implies that
                  stability is algorithmically decidable for
                  $\{\mathrm{S}_1,\mathrm{S}_2\}$.",
}

@ARTICLE{DHX:SIAMJCO08,
 author        = "Dai, Xiongping and Huang, Yu and Xiao, Mingqing",
 title         = "Almost sure stability of discrete-time switched linear
                  systems: a topological point of view",
 journal       = "SIAM J. Control Optim.",
 fjournal      = "SIAM Journal on Control and Optimization",
 year          = "2008",
 volume        = "47",
 number        = "4",
 pages         = "2137--2156",
 coden         = "SJCODC",
 issn          = "0363-0129",
 mrclass       = "93E15 (37B10 37H15 93C55)",
 mrnumber      = "2421343 (2009i:93134)",
 mrreviewer    = "Sylvie Ruette",
 zblnumber     = "1165.93030",
 doi           = "10.1137/070699676",
 url           = "https://epubs.siam.org/doi/10.1137/070699676",
 annote        = "We study the stability of discrete-time switched linear
                  systems via symbolic topology formulation and the
                  multiplicative ergodic theorem. A sufficient and necessary
                  condition for $\mu_A$-almost sure stability is derived,
                  where $\mu_A$ is the Parry measure of the topological
                  Markov chain with a prescribed transition $(0,1)$-matrix
                  $A$. The obtained $\mu_A$-almost sure stability is
                  invariant under small perturbations of the system. The
                  topological description of stable processes of switched
                  linear systems in terms of Hausdorff dimension is given,
                  and it is shown that our approach captures the maximal set
                  of stable processes for linear switched systems. The
                  obtained results cover the stochastic Markov jump linear
                  systems, where the measure is the natural Markov measure
                  defined by the transition probability matrix. Two examples
                  are provided to illustrate the theoretical outcomes of the
                  paper.",
}

@INPROCEEDINGS{DHX:CDC08,
 author        = "Dai, Xiongping and Huang, Yu and Xiao, Mingqing",
 title         = "Topological formulation of discrete-time switched linear
                  systems and almost sure stability",
 booktitle     = "47th IEEE Conference on Decision and Control (CDC), 9-11
                  Dec.",
 publisher     = "IEEE",
 address       = "Cuncun",
 year          = "2008",
 pages         = "965--970",
 doi           = "10.1109/CDC.2008.4738721",
 annote        = "In this paper, we study the stability of discrete-time
                  switched linear systems via symbolic topology formulation
                  and the multiplicative ergodic theorem. A sufficient and
                  necessary condition for $\mu_A$-almost sure stability is
                  derived, where $\mu_A$ is the Parry measure of the
                  topological Markov chain with a prescribed transition
                  $(0,1)$-matrix $A$. The obtained $\mu_A$-almost sure
                  stability is invariant under small perturbations of the
                  system. The topological description of stable processes of
                  switched linear systems in terms of Hausdorff dimension is
                  given, and it is shown that our approach captures the
                  maximal set of stable processes for linear switched
                  systems. The obtained results cover the stochastic Markov
                  jump linear systems, where the measure is the natural
                  Markov measure defined by the transition probability
                  matrix.",
}

@MISC{DHX:ArXiv11-1,
 author        = "Dai, Xiongping and Huang, Yu and Xiao, Mingqing",
 title         = "Pointwise Stabilization of Discrete-time Stationary
                  Matrix-valued {M}arkovian Chains",
 howpublished  = "ArXiv.org e-Print archive",
 year          = "2011",
 month         = jul,
 eprinttype    = "arXiv",
 eprint        = "1107.0132",
 doi           = "10.48550/arXiv.1107.0132",
 annote        = "We study the pointwise stabilizability of a discretetime,
                  time-homogeneous, and stationary Markovian jump linear
                  system. By using measure theory, ergodic theory and a
                  splitting theorem of state space we show in a relatively
                  simple way that if the system is essentially
                  product-bounded, then it is pointwise convergent if and
                  only if it is pointwise exponentially convergent, which
                  provides an important characteristic of pointwise
                  convergence under the framework of symbolic dynamics.",
}

@ARTICLE{DHX:ERA11,
 author        = "Dai, Xiongping and Huang, Yu and Xiao, Mingqing",
 title         = "Realization of joint spectral radius via ergodic theory",
 journal       = "Electron. Res. Announc. Math. Sci.",
 fjournal      = "Electronic Research Announcements in Mathematical Sciences",
 year          = "2011",
 volume        = "18",
 pages         = "22--30",
 issn          = "1935-9179",
 mrclass       = "37A30 (15A18 15A60 37H15)",
 mrnumber      = "2817401 (2012f:37007)",
 mrreviewer    = "Ian Melbourne",
 zblnumber     = "1218.15004",
 doi           = "10.3934/era.2011.18.22",
 url           = "https://www.aimsciences.org/journals/displayArticles.jsp?paperID=6243",
 annote        = "Based on the classic multiplicative ergodic theorem and the
                  semi-uniform subadditive ergodic theorem, we show that
                  \emph{there always exists at least one ergodic Borel
                  probability measure such that the joint spectral radius of
                  a finite set of square matrices of the same size can be
                  realized almost everywhere with respect to this Borel
                  probability measure.} The existence of at least one ergodic
                  Borel probability measure, in the context of the joint
                  spectral radius problem, is obtained in a general
                  setting.",
}

@ARTICLE{DHX:AUT11,
 author        = "Dai, Xiongping and Huang, Yu and Xiao, Mingqing",
 title         = "Periodically switched stability induces exponential
                  stability of discrete-time linear switched systems in the
                  sense of {M}arkovian probabilities",
 journal       = "Automatica J. IFAC",
 fjournal      = "Automatica. A Journal of IFAC, the International Federation
                  of Automatic Control",
 year          = "2011",
 volume        = "47",
 number        = "7",
 pages         = "1512--1519",
 coden         = "ATCAA9",
 issn          = "0005-1098",
 mrclass       = "37N35 (93E15)",
 mrnumber      = "2889251",
 zblnumber     = "1219.93142",
 doi           = "10.1016/j.automatica.2011.02.034",
 url           = "https://www.sciencedirect.com/science/article/abs/pii/S000510981100135X",
 annote        = "The conjecture that periodically switched stability implies
                  absolute asymptotic stability of random infinite products
                  of a finite set of square matrices, has recently been
                  disproved under the guise of the finiteness conjecture. In
                  this paper, we show that this conjecture holds in terms of
                  Markovian probabilities. More specifically, let $S_{k}\in
                  \mathbb{C}^{n\times n}$, $1\le k\le K$, be arbitrarily
                  given $K$ matrices and $\Sigma^{+}_{K} =
                  \{(k_{j})^{\infty}_{j=1}~ |~ 1\le k_j\le K~ \textrm{for
                  each}~j\ge 1\}$, where $n, K \ge 2$. Then we study the
                  exponential stability of the following discrete-time
                  switched dynamics $S$: \[ x_j = S_{k_j}\cdots S_{k_1}
                  x_0,\quad j\ge 1~\textrm{and}~ x_0\in \mathbb{C}^n\] where
                  $(k_{j})^{\infty}_{j=1}\in \Sigma^{+}_{K}$ can be an
                  arbitrary switching sequence. For a probability row-vector
                  $\mathbf{p} = (p_1,\ldots, p_K )\in \mathbb{R}^{K}$ and an
                  irreducible Markov transition matrix $\mathsf{P}\in
                  \mathbb{R}^{K\times K}$ with $\mathbf{p}\mathsf{P} =
                  \mathbf{p}$, we denote by $\mu_{\mathbf{p},\mathsf{P}}$ the
                  Markovian probability on $\Sigma^{+}_{K}$ corresponding to
                  $(\mathbf{p},\mathsf{P})$. By using symbolic dynamics and
                  ergodic-theoretic approaches, we show that, if $S$
                  possesses the periodically switched stability then, (i) it
                  is exponentially stable
                  $\mu_{\mathbf{p},\mathsf{P}}$-almost surely; (ii) the set
                  of stable switching sequences $(k_{j})^{\infty}_{j=1}\in
                  \Sigma^{+}_{K}$ has the same Hausdorff dimension as
                  $\Sigma^{+}_{K}$. Thus, the periodically switched stability
                  of a discrete-time linear switched dynamics implies that
                  the system is exponentially stable for ``almost'' all
                  switching sequences.",
}

@MISC{DHX:ArXiv11-2,
 author        = "Dai, Xiongping and Huang, Yu and Xiao, Mingqing",
 title         = "Stability Criteria via Common Non-strict {L}yapunov Matrix
                  for Discrete-time Linear Switched Systems",
 howpublished  = "ArXiv.org e-Print archive",
 year          = "2011",
 month         = aug,
 eprinttype    = "arXiv",
 eprint        = "1108.0239",
 doi           = "10.48550/arXiv.1108.0239",
 annote        = "Let $\boldsymbol{S}=\{S_1,S_2\}\subset\mathbb{R}^{d\times
                  d}$ have a common, but not necessarily strict, Lyapunov
                  matrix (i.e. there exists a symmetric positive-definite
                  matrix $P$ such that $P-S_k^TPS_k\ge0$ for $k=1,2$). Based
                  on a splitting theorem of the state space
                  $\mathbb{R}^d$~\cite{DHX:ArXiv11-1}, we establish several
                  stability criteria for the discrete-time linear switched
                  dynamics \[ x_n=S_{\sigma_{\!n}}\cdots
                  S_{\sigma_{\!1}}(x_0),\quad x_0\in\mathbb{R}^d\textrm{ and
                  }n\ge1 \] governed by the switching signal
                  $\sigma\colon\mathbb{N}\rightarrow\{1,2\}$. More
                  specifically, let $\rho(A)$ stand for the spectral radius
                  of a matrix $A\in\mathbb{R}^{d\times d}$, then the outline
                  of results obtained in this paper
                  are:\begin{enumerate}\item[(1)] \emph{For the case $d=2$,
                  $\boldsymbol{S}$ is absolutely stable (i.e.,
                  $\|S_{\sigma_{\!n}}\cdots S_{\sigma_{\!1}}\|\to0$ driven by
                  all switching signals $\sigma$) if and only if
                  $\rho(S_1),\rho(S_2)$ and $\rho(S_1S_2)$ all are less than
                  $1$;}\item[(2)] \emph{For the case $d=3$, $\boldsymbol{S}$
                  is absolutely stable if and only if $\rho(A)<1\;\forall
                  A\in\{S_1,S_2\}^\ell$ for $\ell=1,2,3,4,5,6$, and
                  $8$.}\end{enumerate} This further implies that for any
                  $\boldsymbol{S}=\{S_1,S_2\}\subset\mathbb{R}^{d\times d}$
                  with the generalized spectral radius
                  $\rho(\boldsymbol{S})=1$ where $d=2$ or $3$, if
                  $\boldsymbol{S}$ has a common, but not strict in general,
                  Lyapunov matrix, then $\boldsymbol{S}$ possesses the
                  spectral finiteness property.",
}

@MISC{DHX:ArXiv13-1,
 author        = "Dai, Xiongping and Huang, Yu and Xiao, Mingqing",
 title         = "Spectral Finiteness Property and Splitting of State Space
                  for Matrix-Valued Cocycles",
 howpublished  = "ArXiv.org e-Print archive",
 year          = "2013",
 month         = aug,
 eprinttype    = "arXiv",
 eprint        = "1308.6111",
 doi           = "10.48550/arXiv.1308.6111",
 annote        = "By proving and using a splitting of state space without the
                  uniform product boundedness condition, it is studied the
                  effectiveness of the computation of the joint spectral
                  radius for a linear cocycle driven by a discrete-time
                  continuous semiflow. Our result may be applied to Markovian
                  jump linear systems.",
}

@ARTICLE{DHX:PAMS13,
 author        = "Dai, Xiongping and Huang, Yu and Xiao, MingQing",
 title         = "Extremal ergodic measures and the finiteness property of
                  matrix semigroups",
 journal       = "Proc. Amer. Math. Soc.",
 fjournal      = "Proceedings of the American Mathematical Society",
 year          = "2013",
 volume        = "141",
 number        = "2",
 pages         = "393--401",
 eprinttype    = "arXiv",
 eprint        = "1107.0123",
 coden         = "PAMYAR",
 issn          = "0002-9939",
 mrclass       = "15A60 (15A30 15B52)",
 mrnumber      = "2996944",
 mrreviewer    = "Sho Matsumoto",
 zblnumber     = "06141459",
 doi           = "10.1090/S0002-9939-2012-11330-9",
 url           = "https://www.ams.org/journals/proc/2013-141-02/S0002-9939-2012-11330-9/",
 annote        = "Let $\mathbf{S}=\{S_1,\ldots,S_K\}$ be a finite set of,
                  complex, $d\times d$ matrices and, $\varSigma_{\!K}^+$ the
                  compact space of all one-sided infinite sequences
                  $i_{.}\colon\mathbb{N}\rightarrow\{1,\ldots,K\}$. An
                  ergodic probability $\mu_*$ of the Markov shift
                  $\theta\colon\varSigma_{\!K}^+\rightarrow\varSigma_{\!K}^+;\
                  i_{.}\mapsto i_{.+1}$, is called ``extremal'' for
                  $\mathbf{S}$, if
                  ${\rho}(\mathbf{S})=\lim_{n\to\infty}\sqrt[n]{\|{S_{i_1}\cdots
                  S_{i_n}}\|}$ holds for $\mu_*$-a.e.
                  $i_{.}\in\varSigma_{\!K}^+$, where $\rho(\mathbf{S})$
                  denotes the generalized/joint spectral radius of
                  $\mathbf{S}$. Using extremal norm and Kingman subadditive
                  ergodic theorem, it is shown:\par \textbf{Theorem.}
                  \emph{$\mathbf{S}$ has the spectral finiteness property
                  (i.e. $\rho(\mathbf{S})=\sqrt[n]{\rho(S_{i_1}\cdots
                  S_{i_n})}$ for some finite-length word $(i_1,\ldots,i_n)$)
                  if and only if for some extremal measure $\mu_*$ of
                  $\mathbf{S}$, it has at least one periodic density point
                  $i_{.}\in\varSigma_{\!K}^+$.}",
}

@ARTICLE{DHX:IEEETAC15,
 author        = "Dai, Xiongping and Huang, Yu and Xiao, Mingqing",
 title         = "Pointwise Stability of Discrete-time Homogeneous
                  Matrix-valued {M}arkovian Processes",
 journal       = "IEEE Trans. Automat. Control",
 fjournal      = "Institute of Electrical and Electronics Engineers.
                  Transactions on Automatic Control",
 year          = "2015",
 volume        = "60",
 number        = "7",
 pages         = "1898--1903",
 issn          = "0018-9286",
 mrclass       = "60J75 (93E15)",
 mrnumber      = "3365076",
 zblnumber     = "1360.93736",
 doi           = "10.1109/TAC.2014.2361594",
 url           = "https://ieeexplore.ieee.org/document/6917019",
 annote        = "In this note we study the pointwise stability of a
                  discrete-time, matrix-valued, and stationary Markovian jump
                  linear system. When the system is restricted to a linear
                  subspace, we show that it is pointwise convergent if and
                  only if it is pointwise exponentially convergent.",
}

@ARTICLE{DK:INFOPROC11:e,
 author        = "Dai, Xiongping and Kozyakin, Victor",
 title         = "Finiteness Property of a Bounded Set of Matrices with
                  Uniformly Sub-Peripheral Spectrum",
 journal       = "J. Commun. Technol. Electron.",
 fjournal      = "Journal of Communications Technology and Electronics",
 year          = "2011",
 volume        = "56",
 number        = "12",
 pages         = "1564--1569",
 eprinttype    = "arXiv",
 eprint        = "1106.2298",
 doi           = "10.1134/S1064226911120096",
 url           = "https://link.springer.com/article/10.1134/S1064226911120096",
 annote        = "In the paper, a simple condition guaranteing the finiteness,
                  property, for a bounded set $\boldsymbol{S}=\{S_k\}_{k\in
                  K}$ of real or complex $d\times d$ matrices, is presented.
                  It is shown that existence of a sequence of matrix products
                  $\boldsymbol{S}_{\!{\sigma(n_{\ell})}}$ of length $n_\ell$
                  for $\boldsymbol{S}$ with $n_{\ell}\to\infty$ such that the
                  spectrum of each matrix
                  $\boldsymbol{S}_{\!{\sigma(n_{\ell})}}$ is uniformly
                  sub-peripheral and \[
                  {\rho}({\boldsymbol{S}}):=\sup_{n\ge1}\sup_{i_1,\ldots,i_n\in
                  K} \sqrt[n]{{\rho(S_{i_1}\cdots
                  S_{i_n})}}=\lim_{\ell\to+\infty}
                  \sqrt[\uproot{3}n_{\ell}]{\rho(\boldsymbol{S}_{\!{\sigma(n_{\ell})}})},
                  \] guarantees the spectral finiteness property for
                  $\boldsymbol{S}$.",
}

@ARTICLE{DaubLag:SIAMMAN91,
 author        = "Daubechies, Ingrid and Lagarias, Jeffrey C.",
 authauthor    = "Daubechies, Ingrid and Lagarias, Jeffrey",
 title         = "Two-scale difference equations. {I}. {E}xistence and global
                  regularity of solutions",
 journal       = "SIAM J. Math. Anal.",
 fjournal      = "SIAM Journal on Mathematical Analysis",
 year          = "1991",
 volume        = "22",
 number        = "5",
 pages         = "1388--1410",
 coden         = "SJMAAH",
 issn          = "0036-1410",
 mrclass       = "39A10 (42C05)",
 mrnumber      = "1112515 (92d:39001)",
 mrreviewer    = "Evangelos Ifantis",
 zblnumber     = "0763.42018",
 zblreviewer   = "R. Vaillancourt (Ottawa)",
 doi           = "10.1137/0522089",
 url           = "https://epubs.siam.org/doi/10.1137/0522089",
 annote        = "A two-scale difference equation is a functional equation of
                  the form $f(x) = \sum_{n = 0}^N c_n f(\alpha x - \beta_n)$,
                  where $\alpha>1$ and $\beta_0 < \beta_1<\cdots<\beta_n$,
                  are real constants, and $c_n$ are complex constants.
                  Solutions of such equations arise in spline theory, in
                  interpolation schemes for constructing curves, in
                  constructing wavelets of compact support, in constructing
                  fractals, and in probability theory. This paper studies the
                  existence and uniqueness of $L^1$-solutions to such
                  equations. In particular, it characterizes $L^1$-solutions
                  having compact support. A time-domain method is introduced
                  for studying the special case of such equations where
                  $\{{\alpha,\beta_0,\ldots,\beta_n}\}$ are integers, which
                  are called lattice two-scale difference equations. It is
                  shown that if a lattice two-scale difference equation has a
                  compactly supported solution in $C^m (\mathbb{R})$, then
                  $m<{{(\beta_n - \beta_0 )} / {(\alpha - 1)}} - 1$.\par Two
                  examples are considered: the de Rham continuous
                  nowhere-differentiable function [communicated by
                  \emph{Y.~Meyer} (1987)] and the Lagrange interpolation
                  functions of Deslauriers and Dubuc [\emph{G.~Deslauriers}
                  and \emph{S.~Dubuc}, Constructive Approximation 5, No. 1,
                  49--68 (1989)].",
}

@ARTICLE{DaubLag:SIAMMAN92,
 author        = "Daubechies, Ingrid and Lagarias, Jeffrey C.",
 authauthor    = "Daubechies, Ingrid and Lagarias, Jeffrey",
 title         = "Two-scale difference equations. {II}. {L}ocal regularity,
                  infinite products of matrices and fractals",
 journal       = "SIAM J. Math. Anal.",
 fjournal      = "SIAM Journal on Mathematical Analysis",
 year          = "1992",
 volume        = "23",
 number        = "4",
 pages         = "1031--1079",
 coden         = "SJMAAH",
 issn          = "0036-1410",
 mrclass       = "39A10",
 mrnumber      = "1166574 (93g:39001)",
 mrreviewer    = "Evangelos Ifantis",
 zblnumber     = "0788.42013",
 zblreviewer   = "R. Vaillancourt (Ottawa)",
 doi           = "10.1137/0523059",
 url           = "https://epubs.siam.org/doi/10.1137/05230591",
 annote        = "This paper studies solutions of the functional equation
                  \[f(x) = \sum_{n = 0}^N {c_n f(kx - n),} \] where $k \ge 2$
                  is an integer, and $\sum\nolimits_{n = 0}^N {c_n = k}$.
                  Part I showed that equations of this type have at most one
                  $L^1$-solution up to a multiplicative constant, which
                  necessarily has compact support in $[0,{N / {k - 1}}]$.
                  This paper gives a time-domain representation for such a
                  function $f(x)$ (if it exists) in terms of infinite
                  products of matrices (that vary as x varies). Sufficient
                  conditions are given on $\{{c_n}\}$ for a continuous
                  nonzero $L^1$-solution to exist. Additional conditions
                  sufficient to guarantee $f \in C^r$ are also given. The
                  infinite matrix product representations is used to bound
                  from below the degree of regularity of such an
                  $L^1$-solution and to estimate the H{\"o}lder exponent of
                  continuity of the highest-order well-defined derivative of
                  $f(x)$. Such solutions $f(x)$ are often smoother at some
                  points than others. For certain $f(x)$ a hierarchy of
                  fractal sets in $\mathbb{R}$ corresponding to different
                  H{\"o}lder exponents of continuity for $f(x)$ is
                  described.",
}

@ARTICLE{DaubLag:LAA92,
 author        = "Daubechies, Ingrid and Lagarias, Jeffrey C.",
 authauthor    = "Daubechies, Ingrid and Lagarias, Jeffrey",
 title         = "Sets of matrices all infinite products of which converge",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "1992",
 volume        = "161",
 pages         = "227--263",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15A18 (15A51 15A52 39A10 40A20 60J10)",
 mrnumber      = "1142737 (93f:15006)",
 mrreviewer    = "James Allen Fill",
 zblnumber     = "0746.15015",
 zblreviewer   = "R. Covaci (Cluj-Napoca)",
 doi           = "10.1016/0024-3795(92)90012-Y",
 url           = "https://www.sciencedirect.com/science/article/pii/002437959290012Y",
 annote        = "An infinite product $\prod_{i=1}^{\infty}\mathsf{M}_{i}$ of
                  matrices converges (on the right) if
                  $\lim_{i\to\infty}\mathsf{M}_{1}\cdots\mathsf{M}_{i}$
                  exists. A set $\Sigma=\{\mathsf{A}_{i}:i\ge 1\}$ of
                  $n\times n$ matrices is called an \emph{RCP set}
                  (right-convergent product set) if all infinite products
                  with each element drawn from $\Sigma$ converge. Such sets
                  of matrices arise in constructing self-similar objects like
                  von Koch's snowflake curve, in various interpolation
                  schemes, in constructing wavelets of compact support, and
                  in studying nonhomogeneous Markov chains. This paper gives
                  necessary conditions and also some sufficient conditions
                  for a set $\Sigma$ to be an RCP set. These are conditions
                  on the eigenvalues and left eigenspaces of matrices in
                  $\Sigma$ and finite products of these matrices. Necessary
                  and sufficient conditions are given for a finite set
                  $\Sigma$ to be an RCP set having a limit function
                  $\mathsf{M}_{\Sigma}(d)={}
                  \prod_{i=1}^{\infty}\mathsf{A}_{d_{i}}$, where
                  $\mathsf{d}=(d_1,\ldots,d_n,\ldots)$, which is a continuous
                  function on the space of all sequences $\mathsf{d}$ with
                  the sequence topology. Finite RCP sets of column-stochastic
                  matrices are completely characterized. Some results are
                  given on the problem of algorithmically deciding if a given
                  set $\Sigma$ is an RCP set.",
}

@ARTICLE{DaubLag:LAA01,
 author        = "Daubechies, Ingrid and Lagarias, Jeffrey C.",
 authauthor    = "Daubechies, Ingrid and Lagarias, Jeffrey",
 title         = "Corrigendum/addendum to: ``{S}ets of matrices all infinite
                  products of which converge'' [{L}inear {A}lgebra {A}ppl.\
                  {\bf 161} (1992), 227--263; {MR}1142737 (93f:15006)]",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2001",
 volume        = "327",
 number        = "1-3",
 pages         = "69--83",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15A18 (15A51 15A52 39A10 40A20 60J10)",
 mrnumber      = "1823340 (2002b:15010)",
 zblnumber     = "0978.15024",
 zblreviewer   = "Rodica Covaci (Cluj-Napoca)",
 doi           = "10.1016/S0024-3795(00)00314-1",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379500003141",
 annote        = "This corrigendum/addendum supplies corrected statements and
                  proofs of some results in the paper~\cite{DaubLag:LAA92}.
                  These results concern special kinds of bounded semigroups
                  of matrices. It also reports on progress on the topics of
                  this paper made in the last eight years.",
}

@INPROCEEDINGS{DGM:LIPICS17,
 author        = "Daviaud, Laure and Guillon, Pierre and Merlet, Glenn",
 editor        = "Larsen, Kim G. and Bodlaender, Hans L. and Raskin,
                  Jean-Francois",
 title         = "Comparison of Max-Plus Automata and Joint Spectral Radius of
                  Tropical Matrices",
 booktitle     = "42nd International Symposium on Mathematical Foundations of
                  Computer Science (MFCS 2017)",
 series        = "Leibniz International Proceedings in Informatics (LIPIcs)",
 publisher     = "Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik",
 address       = "Dagstuhl, Germany",
 year          = "2017",
 volume        = "83",
 pages         = "19:1--19:14",
 eprinttype    = "arXiv",
 eprint        = "1612.02647",
 hal_id        = "hal-01479294",
 hal_version   = "v1",
 isbn          = "978-3-95977-046-0",
 issn          = "1868-8969",
 mrnumber      = "3755312",
 zblnumber     = "1441.68119",
 doi           = "10.4230/LIPIcs.MFCS.2017.19",
 url           = "https://drops.dagstuhl.de/opus/volltexte/2017/8105/",
 annote        = "Weighted automata over the tropical semiring
                  $\mathbb{Z}_{\max} = (\mathbb{Z}\cup\{-\infty\},\max,+)$
                  are closely related to finitely generated semigroups of
                  matrices over $\mathbb{Z}_{\max}$. In this paper, we use
                  results in automata theory to study two quantities
                  associated with sets of matrices: the joint spectral radius
                  and the ultimate rank. We prove that these two quantities
                  are not computable over the tropical semiring, i.e. there
                  is no algorithm that takes as input a finite set of
                  matrices $\Gamma$ and provides as output the joint spectral
                  radius (resp. the ultimate rank) of $\Gamma$. On the other
                  hand, we prove that the joint spectral radius is
                  nevertheless approximable and we exhibit restricted cases
                  in which the joint spectral radius and the ultimate rank
                  are computable. To reach this aim, we study the problem of
                  comparing functions computed by weighted automata over the
                  tropical semiring. This problem is known to be undecidable
                  and we prove that it remains undecidable in some specific
                  subclasses of automata.",
}

@ARTICLE{DekKui:JEMS11,
 author        = "Dekking, F. Michel and Kuijvenhoven, Bram",
 authauthor    = "Dekking, Michel and Kuijvenhoven, Bram",
 title         = "Differences of random {C}antor sets and lower spectral
                  radii",
 journal       = "J. Eur. Math. Soc. (JEMS)",
 fjournal      = "Journal of the European Mathematical Society (JEMS)",
 year          = "2011",
 volume        = "13",
 number        = "3",
 pages         = "733--760",
 eprinttype    = "arXiv",
 eprint        = "0811.0525",
 issn          = "1435-9855",
 mrclass       = "28A80 (28A78 60D05)",
 mrnumber      = "2781931 (2012f:28011)",
 mrreviewer    = "Artemi Berlinkov",
 zblnumber     = "05873843",
 zblreviewer   = "Yuang-Ling Ye (Guangzhou)",
 doi           = "10.4171/JEMS/266",
 url           = "https://www.ems-ph.org/journals/show_abstract.php?issn=1435-9855&vol=13&iss=3&rank=7",
 annote        = "We investigate the question under which conditions the
                  algebraic difference between two independent random Cantor
                  sets $C_1$ and $C_2$ almost surely contains an interval,
                  and when not. The natural condition is whether the sum $d_1
                  +d_2$ of the Hausdorff dimensions of the sets is smaller
                  (no interval) or larger (an interval) than $1$. Palis
                  conjectured that generically it should be true that $d_1
                  +d_2 >1$ should imply that $C_1 -C_2$ contains an interval.
                  We prove that for $2$-adic random Cantor sets generated by
                  a vector of probabilities $(p_0 ,p_1)$ the interior of the
                  region where the Palis conjecture does not hold is given by
                  those $p_0 ,p_1$ which satisfy $p_0 +p_1 >\sqrt{2}$ and
                  $p_0 p_1 (1+p_{0}^{2} +p_{1}^{2})<1$. We furthermore prove
                  a general result which characterizes the interval/no
                  interval property in terms of the lower spectral radius of
                  a set of $2\times 2$ matrices.",
}

@ARTICLE{DRD:IEEETAC18,
 author        = "Della Rossa, F. and Dercole, F.",
 title         = "Tree-based algorithms for the stability of discrete-time
                  switched linear systems under arbitrary and constrained
                  switching",
 journal       = "IEEE Trans. Autom. Control",
 fjournal      = "IEEE Transactions on Automatic Control",
 year          = "2019",
 volume        = "64",
 number        = "9",
 pages         = "3823--3830",
 issn          = "0018-9286",
 mrclass       = "93D20 (05C90 93B25)",
 mrnumber      = "4003174",
 zblnumber     = "07158456",
 doi           = "10.1109/TAC.2018.2887142",
 url           = "https://ieeexplore.ieee.org/document/8579233",
 annote        = "We present a direct approach to study the stability of
                  discrete-time switched linear systems that can be applied
                  to arbitrary switching, as well as when switching is
                  constrained by a switching automaton. We explore the tree
                  of possible matrix products, by pruning the subtrees rooted
                  at contractions and looking for unstable repeatable
                  products. Generically, this simple strategy either
                  terminates with all contracting leafs---showing the
                  system's asymptotic stability---or finds the shortest
                  unstable and repeatable matrix product. Although it behaves
                  in the worst-case as the exhaustive search, we show that
                  its performance is greatly enhanced by measuring
                  contractiveness w.r.t.~sum-of-squares polynomial norms,
                  optimized to minimize the largest expansion among the
                  system's modes.",
}

@ARTICLE{DJM:SCL20,
 author        = "Dettmann, Carl P. and Jungers, Raphael M. and Mason, Paolo",
 authauthor    = "Carl Dettmann and Raphael Jungers and Paolo Mason",
 title         = "Lower bounds and dense discontinuity phenomena for the
                  stabilizability radius of linear switched systems",
 journal       = "Systems Control Lett.",
 fjournal      = "Systems \& Control Letters",
 year          = "2020",
 volume        = "142",
 pages         = "104737, 6",
 eprinttype    = "arXiv",
 eprint        = "2002.10369",
 issn          = "0167-6911",
 mrclass       = "93D09 (15A60 93B25)",
 mrnumber      = "4120003",
 zblnumber     = "1451.93269",
 doi           = "10.1016/j.sysconle.2020.104737",
 url           = "https://www.sciencedirect.com/science/article/abs/pii/S0167691120301183",
 annote        = "We investigate the stabilizability of discrete-time linear,
                  switched systems, when the sole control action of the
                  controller is the switching signal, and when the controller
                  has access to the state of the system in real time. Despite
                  their importance in many control settings, no algorithm is
                  known that allows to decide the stabilizability of such
                  systems, and very simple examples have been known for long,
                  for which the stabilizability question is open.\par We
                  provide {new results} allowing us to bound the so-called
                  stabilizability radius, which characterizes the
                  stabilizability property of discrete-time linear switched
                  systems. These results allous to improve significantly the
                  computation of the stabilizability radius for the
                  above-mentioned examples. As a by-product, we exhibit a
                  discontinuity property for this problem, which brings
                  theoretical understanding of its complexity.",
}

@ARTICLE{DaiOp:AIT01:e,
 author        = "Diamond, P. and Opojtsev, V. I.",
 authauthor    = "Diamond, Phil and Opojtsev, V.",
 title         = "Stability of linear difference and differential inclusions",
 journal       = "Autom. Remote Control",
 fjournal      = "Automation and Remote Control",
 year          = "2001",
 volume        = "62",
 number        = "5",
 pages         = "695--703",
 issn          = "0005-1179",
 mrclass       = "34A60 (15A60 34D99 93D99)",
 mrnumber      = "1852745 (2002k:34026)",
 mrreviewer    = "Nikolai A. Kudryashov",
 zblnumber     = "1082.34509",
 zblreviewer   = "Sotiris K. Ntouyas (Ioannina)",
 doi           = "10.1023/A:1010258420380",
 url           = "https://link.springer.com/article/10.1023/A:1010258420380",
 annote        = "If $S$ is an $n$-dimensional linear space and $A$ the matrix
                  of a linear transformation $\mathcal{A}: S\to S,$ then the
                  space $S$ can always be imbedded into an $N$-dimensional
                  linear space $E$, $N>n$, such that there exists an
                  extension $\mathcal{B}: E\to E$ of $\mathcal{A}$ with a
                  nonnegative matrix $B$ in some base. This property and some
                  of its applications are the subject of this paper. In
                  particular, the authors study the extensions $B$ of the
                  matrix $A$ for which the spectral radius $\rho(B)$ is close
                  to $\rho(A)$. The inclusion $x^{k+1}\in \mathbb{A} x^{k}$
                  is also considered, where $\mathbb{A}$ is a compact set of
                  matrices, and it is proved that this is asymptotically
                  stable if and only if $\mathbb{A}$ can be extended to a set
                  $\mathbb{B}$ of nonnegative matrices $B$ with $\| B\|_{1}$
                  or $\|B\|_{\infty}<1$. Similar results are derived for
                  differential inclusions.",
}

@ARTICLE{Drnov:97,
 author        = "Drnov{\v{s}}ek, Roman",
 title         = "On reducibility of semigroups of compact quasinilpotent
                  operators",
 journal       = "Proc. Amer. Math. Soc.",
 fjournal      = "Proceedings of the American Mathematical Society",
 year          = "1997",
 volume        = "125",
 number        = "8",
 pages         = "2391--2394",
 coden         = "PAMYAR",
 issn          = "0002-9939",
 mrclass       = "47A15 (47D03)",
 mrnumber      = "1422865 (97m:47007)",
 mrreviewer    = "Wac{\l}aw Szyma{\'n}ski",
 zblnumber     = "0883.47001",
 doi           = "10.1090/S0002-9939-97-04108-7",
 url           = "https://www.ams.org/journals/proc/1997-125-08/S0002-9939-97-04108-7/",
 annote        = "The following generalization of Lomonosov's invariant
                  subspace theorem is proved:\par \textbf{Theorem.} \emph{Let
                  $S$ be a multiplicative semigroup of compact operators on a
                  Banach space such that $\hat{r}(S_{1},\ldots,S_{n})=0$ for
                  every finite subset $\{S_{1},\ldots,S_{n}\}$ of $S$, where
                  $\hat{r}$ denotes the Rota--Strang spectral radius. Then
                  $S$ is reducible.}\par This result implies that the
                  following assertions are equivalent:\par (A) \emph{For each
                  infinite-dimensional complex Hilbert space $\mathcal{H}$,
                  every semigroup of compact quasinilpotent operators on
                  $\mathcal{H}$ is reducible.}\par (B) \emph{For every
                  complex Hilbert space $\mathcal{H}$, for every semigroup of
                  compact quasinilpotent operators on $\mathcal{H}$, and for
                  every finite subset $\{S_{1},\ldots,S_{n}\}$ of $S$ it
                  holds that $\hat{r}(S_{1},\ldots,S_{n})=0$.}\par The
                  question whether the assertion (A) is true was considered
                  by Nordgren, Radjavi and Rosenthal in 1984, and it seems to
                  be still open.",
}

@ARTICLE{Drnovsek:SM97,
 author        = "Drnov{\v{s}}ek, Roman",
 title         = "An irreducible semigroup of idempotents",
 journal       = "Studia Math.",
 fjournal      = "Studia Mathematica",
 year          = "1997",
 volume        = "125",
 number        = "1",
 pages         = "97--99",
 issn          = "0039-3223",
 mrclass       = "47A15 (20M99 47D03)",
 mrnumber      = "1455626 (98e:47011)",
 mrreviewer    = "P. Rosenthal",
 zblnumber     = "0886.47005",
 doi           = "10.4064/sm-125-1-97-99",
 url           = "https://www.impan.pl/pl/wydawnictwa/czasopisma-i-serie-wydawnicze/studia-mathematica/all/125/1/110020/an-irreducible-semigroup-of-idempotents",
 annote        = "A semigroup of bounded idempotents with no nontrivial closed
                  subspaces is constructed. This answers a question which was
                  open for some time.",
}

@ARTICLE{Drn:IEOT01,
 author        = "Drnov{\v{s}}ek, Roman",
 title         = "Common invariant subspaces for collections of operators",
 journal       = "Integral Equations Operator Theory",
 fjournal      = "Integral Equations and Operator Theory",
 year          = "2001",
 volume        = "39",
 number        = "3",
 pages         = "253--266",
 issn          = "0378-620X",
 mrclass       = "47A15 (47B60 47D03)",
 mrnumber      = "1818060 (2001m:47012)",
 mrreviewer    = "Gordon Wilson MacDonald",
 zblnumber     = "0994.47008",
 zblreviewer   = "Daniel Beltita (Bucure{\c{s}}ti)",
 doi           = "10.1007/BF01332655",
 url           = "https://link.springer.com/article/10.1007/BF01332655",
 annote        = "The paper concerns common invariant subspaces for sets of
                  operators which are finitely quasinilpotent at some vector.
                  A set $\mathcal{C}$ of bounded linear operators on a Banach
                  space $X$ is said to be finitely quasinilpotent at $x_0\in
                  X$ if for every finite subset $\mathcal{F}$ of
                  $\mathcal{C}$ one has
                  \[\limsup_{n\to\infty}\bigl(\sup_{T_1,\ldots,T_n\in\mathcal{F}}
                  |T_1\cdots T_nx_0|\bigr)^{1/n}=0.\]\par \textbf{Theorem
                  3.3.} \emph{If a set $\mathcal{C}$ of operators on $X$ is
                  finitely quasinilpotent at some non-vector and
                  $\mathcal{C}$ contains a non-zero compact operator, then
                  there exists a nontrivial subspace of $X$ which is
                  invariant to each operator belonging either to
                  $\mathcal{C}$ or to the commutant of $\mathcal{C}$.}\par
                  \textbf{Theorem 3.5.} \emph{The same fact is valid in the
                  case when $\mathcal{C}$ ($\ne\{0\}$) is finitely
                  quasinilpotent at some non-zero vector and there exists a
                  non-zero compact operator in the commutant of
                  $\mathcal{C}$.}\par The proof of Theorem 3.3 is based on
                  Hilden's ``ping-pong'' method of proof of the celebrated
                  Lomonosov's invariant subspace theorem, exposed by
                  \emph{A.\,J.~Michaels} [Adv. Math. 25, 56--58 (1977)]. \par
                  The last part of the paper includes versions of the above
                  described results for sets of positive operators on Banach
                  lattices. In particular, one extends a famous theorem of
                  \emph{B.~de\,Pagter} [Math. Z. 192, 149--153 (1986)].",
}

@ARTICLE{Dumas:LAA13,
 author        = "Dumas, Philippe",
 title         = "Joint spectral radius, dilation equations, and asymptotic
                  behavior of radix-rational sequences",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2013",
 volume        = "438",
 number        = "5",
 pages         = "2107--2126",
 issn          = "0024-3795",
 mrclass       = "11A63 (41A60)",
 mrnumber      = "3005279",
 mrreviewer    = "Horst Brunotte",
 zblnumber     = "1368.11005",
 zblreviewer   = "Jean-Paul Allouche",
 doi           = "10.1016/j.laa.2012.10.013",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379512007239",
 annote        = "Radix-rational sequences are solutions of systems of
                  recurrence equations based on the radix representation of
                  the index. For each radix-rational sequence with complex
                  values we provide an asymptotic expansion, essentially in
                  the scale $N^{\alpha}\log^{\ell}N$. The precision of the
                  asymptotic expansion depends on the joint spectral radius
                  of the linear representation of the sequence of first-order
                  differences. The coefficients are H{\"o}lderian functions
                  obtained through some dilation equations, which are usual
                  in the domains of wavelets and refinement schemes. The
                  proofs are ultimately based on elementary linear algebra.",
}

@ARTICLE{Els:LAA95,
 author        = "Elsner, L.",
 title         = "The generalized spectral-radius theorem: an
                  analytic-geometric proof",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "1995",
 volume        = "220",
 pages         = "151--159",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15A18 (15A60)",
 mrnumber      = "1334574 (96e:15010)",
 mrreviewer    = "Zden{\v{e}}k Dost{\'a}l",
 zblnumber     = "0828.15006",
 zblreviewer   = "E. Kreyszig (Ottawa)",
 doi           = "10.1016/0024-3795(93)00320-Y",
 url           = "https://www.sciencedirect.com/science/article/pii/002437959300320Y",
 note          = "Proceedings of the {W}orkshop ``{N}onnegative {M}atrices,
                  {A}pplications and {G}eneralizations'' and the {E}ighth
                  {H}aifa {M}atrix {T}heory {C}onference ({H}aifa, 1993)",
 annote        = "Let $\Sigma$ be a bounded set of complex matrices,
                  $\Sigma^{m} = \{A_{1}\cdots A_{m}: A_{i}\in \Sigma\}$. The
                  generalized spectral-radius theorem states that
                  $\rho(\Sigma) =\hat{\rho}(\Sigma)$, where $\rho(\Sigma)$
                  and $\hat{\rho}(\Sigma)$ are defined as follows:
                  $\rho(\Sigma) =\limsup\rho_{m}(\Sigma)^{1/m}$, where
                  $\rho_{m}(\Sigma) =\sup\{\rho(A): A\in\Sigma^{m}\}$ with
                  $\rho(A)$ the spectral radius; $\hat{\rho}(\Sigma)
                  =\limsup\rho_{m}(\Sigma)^{1/m}$, where
                  $\hat{\rho}_{m}(\Sigma) =\sup\{\|A\|: A\in\Sigma^{m}\}$
                  with $\|~\|$ any matrix norm.\par The theorem in the title
                  was proved for real matrices by \emph{M.\,A.~Berger} and
                  \emph{Y.~Wang}~\cite{BerWang:LAA92}, using results from the
                  theory of rings. In the present paper the theorem is proved
                  by tools from analysis and geometry (convexity) as well as
                  by some elementary results from the Berger--Wang paper,
                  which is in some ways simpler than the first proof by
                  Berger and Wang.",
}

@ARTICLE{EF:LAA97,
 author        = "Elsner, L. and Friedland, S.",
 title         = "Norm conditions for convergence of infinite products",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "1997",
 volume        = "250",
 pages         = "133--142",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15A60",
 mrnumber      = "1420574 (97m:15046)",
 mrreviewer    = "Israel Koltracht",
 zblnumber     = "0868.40001",
 zblreviewer   = "R. J. Tomkins (Regina)",
 doi           = "10.1016/0024-3795(95)00423-8",
 url           = "https://www.sciencedirect.com/science/article/pii/0024379595004238",
 annote        = "A set $\Sigma$ of $n\times n$ complex-valued matrices is
                  defined to be an LCP set if every infinite left product of
                  members of $\Sigma$ is convergent. Necessary and sufficient
                  conditions for a set of matrices to be an LCP set have been
                  provided by \emph{I.~Daubechies} and
                  \emph{J.\,C.~Lagarias}~\cite{DaubLag:LAA92}, but their
                  result involves the use of an infinite number of
                  conditions. Straightforward conditions tantamount to the
                  LCP property are given for a two-member set $\Sigma$ as
                  follows: The set $\Sigma=\{A_1,A_2\}$ is an LCP set if and
                  only if a norm $|\cdot|$ exists on $C^n$ such that, for
                  $i=1,2$, (i) $|A_i|\leq 1$, (ii) the intersection of the
                  spectrum of $A_i$ with the unit circle is either empty or
                  $\{1\}$, and (iii) $|A_1A_2x|=|x|$ when and only when
                  $A_1x=A_2x=x$.",
}

@ARTICLE{EE:PAMS04,
 author        = "Emel'yanov, E. {\relax{}Yu}. and Ercan, Z.",
 authauthor    = "Emel'yanov, E. and Ercan, Z.",
 title         = "A formula for the joint local spectral radius",
 journal       = "Proc. Amer. Math. Soc.",
 fjournal      = "Proceedings of the American Mathematical Society",
 year          = "2004",
 volume        = "132",
 number        = "5",
 pages         = "1449--1451",
 coden         = "PAMYAR",
 issn          = "0002-9939",
 mrclass       = "47A13 (47A11)",
 mrnumber      = "2053352 (2004m:47010)",
 zblnumber     = "1062.47006",
 zblreviewer   = "Andrzej So{\l}tysiak (Pozna{\'n})",
 doi           = "10.1090/S0002-9939-03-07199-5",
 url           = "https://www.ams.org/journals/proc/2004-132-05/S0002-9939-03-07199-5/",
 annote        = "The joint spectral radius for a bounded subset $M$ of the
                  algebra of all bounded linear operators on a Banach space
                  $X$ is defined by the formula $\rho(M)=\limsup_{n\to\infty}
                  \|M^n\|^{1/n}$, where $M^n$ denotes the set of all products
                  $T_1 T_2\cdots T_n$ $(T_j\in M)$ and $\|M^n\|=\sup\{\|T\|:
                  T\in M^n\}$. Its local counterpart is defined as follows:
                  $\rho_x(M)= \limsup_{n\to\infty}\|M^nx\|^{1/n}$, where
                  $x\in X$ and $\|M^nx\|= \sup\{\|Tx\|: T\in M^n\}$.\par The
                  main result of the present paper gives the following
                  formula: \[ \rho_x(M)=\sup\limits_{f\in
                  X^\ast}\limsup\limits_{n\to\infty} |f\circ M^n(x)|^{1/n},
                  \] where $|f\circ M^n(x)|=\sup\{| f(Tx)|: T\in M^n\}$. As a
                  corollary to this result, the analogous formula for the
                  joint spectral radius of a bounded set in a Banach algebra
                  is obtained.",
}

@MISC{EppWirth:ArXiv23,
 author        = "Epperlein, Jeremias and Wirth, Fabian",
 title         = "The joint spectral radius is pointwise {H}{\"o}lder
                  continuous",
 howpublished  = "ArXiv.org e-Print archive",
 year          = "2023",
 month         = nov,
 eprinttype    = "arXiv",
 eprint        = "2311.18633",
 doi           = "10.48550/arXiv.2311.18633",
 annote        = "We show that the joint spectral radius is pointwise
                  H{\"o}lder continuous. In addition, the joint spectral
                  radius is locally H{\"o}lder continuous for
                  $\varepsilon$-inflations. In the two-dimensional case,
                  local H{\"o}lder continuity holds on the matrix sets with
                  positive joint spectral radius.",
}

@ARTICLE{FayKri:N08,
 author        = "Fayad, Bassam and Krikorian, Rapha{\"e}l",
 title         = "Exponential growth of product of matrices in
                  {$\textrm{SL}(2,\mathbb{R})$}",
 journal       = "Nonlinearity",
 fjournal      = "Nonlinearity",
 year          = "2008",
 volume        = "21",
 number        = "2",
 pages         = "319--323",
 coden         = "NONLE5",
 issn          = "0951-7715",
 mrclass       = "37C85 (37A99 37H15)",
 mrnumber      = "2384551 (2009d:37041)",
 zblnumber     = "1142.37024",
 doi           = "10.1088/0951-7715/21/2/007",
 url           = "https://iopscience.iop.org/article/10.1088/0951-7715/21/2/007",
 annote        = "In this paper the exponential growth of products of two
                  matrices $A,B \in {\textrm{SL}(2,\mathbb{R})}$ is
                  investigated. It is proved, assuming $A$ is a fixed
                  hyperbolic matrix, that for Lebesgue almost every $B$,
                  products of length $n$ involving less than $n\alpha$, $0
                  \le\alpha< 1/2$ matrices B are uniformly bounded from below
                  by $\gamma n$ for some $\gamma > 1$.",
}

@ARTICLE{FLS:ETDS19,
 author        = "Feng, De-Jun and Lo, Chiu-Hong and Shen, Shuang",
 title         = "Uniformity of {L}yapunov exponents for non-invertible
                  matrices",
 journal       = "Ergodic Theory Dynam. Systems",
 publisher     = "Cambridge University Press",
 year          = "2020",
 volume        = "40",
 number        = "9",
 pages         = "2399--2433",
 eprinttype    = "arXiv",
 eprint        = "1702.07251",
 issn          = "0143-3857",
 mrclass       = "37H15 (28A80 37D25)",
 mrnumber      = "4130809",
 zblnumber     = "07228221",
 doi           = "10.1017/etds.2019.4",
 url           = "https://dx.doi.org/10.1017/etds.2019.4",
 annote        = "Let $\mathbf{M}=(M_1,\ldots, M_k)$ be a tuple of real
                  $d\times d$ matrices. Under certain irreducibility
                  assumptions, we give checkable criteria for deciding
                  whether $\mathbf{M}$ possesses the following property:
                  there exist two constants $\lambda\in\mathbb{R}$ and $C>0$
                  such that for any $n\in \mathbb{N}$ and any $i_1, \ldots,
                  i_n \in \{1,\ldots, k\}$, either $M_{i_1} \cdots
                  M_{i_n}={\bf 0}$ or $C^{-1} e^{\lambda n} \leq \| M_{i_1}
                  \cdots M_{i_n} \| \leq C e^{\lambda n}$, where $\|\cdot\|$
                  is a matrix norm. The proof is based on symbolic dynamics
                  and the thermodynamic formalism for matrix products. As
                  applications, we are able to check the absolute continuity
                  of a class of overlapping self-similar measures on
                  $\mathbb{R}$, the absolute continuity of certain
                  self-affine measures in $\mathbb{R}^d$ and the dimensional
                  regularity of a class of sofic affine-invariant sets in the
                  plane.\par Besides, we clarify the relations between
                  different properties on Lyapunov exponent of a family of
                  non-negative matrices.",
}

@ARTICLE{ForValt:IEEETAC12,
 author        = "Fornasini, Ettore and Valcher, Maria Elena",
 authauthor    = "Fornasini, Ettore and Valcher, Maria",
 title         = "Stability and stabilizability criteria for discrete-time
                  positive switched systems",
 journal       = "IEEE Trans. Automat. Control",
 fjournal      = "Institute of Electrical and Electronics Engineers.
                  Transactions on Automatic Control",
 year          = "2012",
 volume        = "57",
 number        = "5",
 pages         = "1208--1221",
 coden         = "IETAA9",
 issn          = "0018-9286",
 mrclass       = "93D20 (93C30 93D15)",
 mrnumber      = "2923880",
 mrreviewer    = "Tibor Tak{\'a}cs",
 zblnumber     = "1369.93526",
 doi           = "10.1109/TAC.2011.2173416",
 url           = "https://ieeexplore.ieee.org/document/6060861",
 annote        = "In this paper we consider the class of discrete-time
                  switched systems switching between autonomous positive
                  subsystems. First, sufficient conditions for testing
                  stability, based on the existence of special classes of
                  common Lyapunov functions, are investigated, and these
                  conditions are mutually related, thus proving that if a
                  linear copositive common Lyapunov function can be found,
                  then a quadratic positive definite common function can be
                  found, too, and this latter, in turn, ensures the existence
                  of a quadratic copositive common function. Secondly,
                  stabilizability is introduced and characterized. It is
                  shown that if these systems are stabilizable, they can be
                  stabilized by means of a periodic switching sequence, which
                  asymptotically drives to zero every positive initial state.
                  Conditions for the existence of state-dependent stabilizing
                  switching laws, based on the values of a copositive
                  (linear/ quadratic) Lyapunov function, are investigated and
                  mutually related, too.\par Finally, some properties of the
                  patterns of the stabilizing switching sequences are
                  investigated, and the relationship between a sufficient
                  condition for stabilizability (the existence of a Schur
                  convex combination of the subsystem matrices) and an
                  equivalent condition for stabilizability (the existence of
                  a Schur matrix product of the subsystem matrices) is
                  explored.",
}

@INCOLLECTION{FroSpi01,
 author        = "Frommer, A. and Spiteri, P.",
 title         = "On linear asynchronous iterations when the spectral radius
                  of the modulus matrix is one",
 booktitle     = "Topics in numerical analysis",
 series        = "Comput. Suppl.",
 publisher     = "Springer",
 address       = "Vienna",
 year          = "2001",
 volume        = "15",
 pages         = "91--104",
 mrclass       = "65F10 (65Y05)",
 mrnumber      = "1874506 (2002m:65020)",
 mrreviewer    = "Rafael Bru",
 zblnumber     = "0994.65034",
 zblreviewer   = "Dietrich Braess (Bochum)",
 doi           = "10.1007/978-3-7091-6217-0_8",
 url           = "https://link.springer.com/chapter/10.1007/978-3-7091-6217-0_8",
 annote        = "A classical result on linear asynchronous iterations states
                  that convergence occurs if and only if the spectral radius
                  of the modulus matrix is less than 1. The present paper
                  shows that if one introduces very mild restrictions on the
                  admissible asynchronous processes, one gets convergence for
                  a larger class of matrices for which the spectral radius of
                  the modulus matrix is allowed to be equal to 1. The mild
                  restrictions are virtually always fulfilled in practical
                  implementations. In this manner, our result contributes to
                  the better understanding of the different hypotheses
                  underlying mathematical models for asynchronous
                  iterations.",
}

@ARTICLE{FromSzyld:JCAM00,
 author        = "Frommer, Andreas and Szyld, Daniel B.",
 authauthor    = "Frommer, Andreas and Szyld, Daniel",
 title         = "On asynchronous iterations",
 journal       = "J. Comput. Appl. Math.",
 fjournal      = "Journal of Computational and Applied Mathematics",
 year          = "2000",
 volume        = "123",
 number        = "1-2",
 pages         = "201--216",
 coden         = "JCAMDI",
 issn          = "0377-0427",
 mrclass       = "65F10 (65Y05)",
 mrnumber      = "1798526 (2001j:65052)",
 zblnumber     = "0967.65066",
 zblreviewer   = "Willi Sch{\"o}nauer (Karlsruhe)",
 doi           = "10.1016/S0377-0427(00)00409-X",
 url           = "https://www.sciencedirect.com/science/article/pii/S037704270000409X",
 note          = "Numerical analysis 2000, Vol. III. Linear algebra",
 annote        = "Theory of asynchronous iterations, basically as extensions
                  of synchronous convergence theorems to asynchronous
                  iteration models for shared memory parallel computers.
                  Application to linear systems, nonlinear systems (Newton)
                  and waveform relaxation. However, for practical use
                  ``wasted operations'' as well as synchronization and
                  communication on distributed memory computers would be of
                  interest.",
}

@ARTICLE{Gaubert:IEEETAC95,
 author        = "Gaubert, St{\'e}phane",
 title         = "Performance evaluation of {$(\max,+)$} automata",
 journal       = "IEEE Trans. Automat. Control",
 fjournal      = "Institute of Electrical and Electronics Engineers.
                  Transactions on Automatic Control",
 year          = "1995",
 volume        = "40",
 number        = "12",
 pages         = "2014--2025",
 issn          = "0018-9286",
 mrclass       = "68Q70 (68Q05 93C83)",
 mrnumber      = "1364950",
 zblnumber     = "0855.93019",
 zblreviewer   = "G. J. Olsder",
 doi           = "10.1109/9.478227",
 url           = "https://ieeexplore.ieee.org/document/478227/",
 annote        = "Automata with multiplicities over the $(\max,+)$ semiring
                  can be used to represent the behavior of timed
                  discrete-event systems. This formalism, which extends both
                  conventional automata and $(\max,+)$ linear
                  representations, covers a class of systems with
                  synchronization phenomena and variable schedules.
                  Performance evaluation is considered in the worst, mean,
                  and optimal cases. A simple algebraic reduction is provided
                  for the worst case. The last two cases are solved for the
                  subclass of deterministic series (recognized by
                  deterministic automata). Deterministic series frequently
                  arise due to the finiteness properties of $(\max,+)$ linear
                  projective semigroups. The mean performance is given by the
                  Kolmogorov equation of a Markov chain. The optimal
                  performance is given by a Hamilton--Jacobi--Bellman
                  equation.",
}

@INPROCEEDINGS{GS:CDC17,
 author        = "Gaubert, St{\'e}phane and Stott, Nikolas",
 title         = "Tropical {K}raus maps for optimal control of switched
                  systems",
 booktitle     = "Proceedings of the 56th Annual Conference on Decision and
                  Control (CDC), 12-15 Dec.",
 publisher     = "IEEE",
 address       = "Melbourne, Victoria, Australia",
 year          = "2017",
 pages         = "1330--1337",
 eprinttype    = "arXiv",
 eprint        = "1706.04471",
 hal_id        = "hal-01674496",
 hal_version   = "v1",
 doi           = "10.1109/CDC.2017.8263839",
 url           = "https://ieeexplore.ieee.org/document/8263839",
 annote        = "Kraus maps (completely positive trace preserving maps) arise
                  classically in quantum information, as they describe the
                  evolution of noncommutative probability measures. We
                  introduce tropical analogues of Kraus maps, obtained by
                  replacing the addition of positive semidefinite matrices by
                  a multivalued supremum with respect to the L{\"o}wner
                  order. We show that non-linear eigenvectors of tropical
                  Kraus maps determine piecewise quadratic approximations of
                  the value functions of switched optimal control problems.
                  This leads to a new approximation method, which we
                  illustrate by two applications: 1) approximating the joint
                  spectral radius, 2) computing approximate solutions of
                  Hamilton--Jacobi PDE arising from a class of switched
                  linear quadratic problems studied previously by McEneaney.
                  We report numerical experiments, indicating a major
                  improvement in terms of scalability by comparison with
                  earlier numerical schemes, owing to the ``LMI-free'' nature
                  of our method.",
}

@ARTICLE{GS:MCRF20,
 author        = "Gaubert, St{\'e}phane and Stott, Nikolas",
 title         = "A convergent hierarchy of non-linear eigenproblems to
                  compute the joint spectral radius of nonnegative matrices",
 journal       = "Mathematical Control \&{} Related Fields",
 year          = "2020",
 volume        = "10",
 number        = "3",
 pages         = "573--590",
 eprinttype    = "arXiv",
 eprint        = "1805.03284",
 hal_id        = "hal-01967552",
 hal_version   = "v1",
 issn          = "2156-8472",
 mrclass       = "15A60 (15A42 47J10 93C30)",
 mrnumber      = "4128840",
 zblnumber     = "07293645",
 doi           = "10.3934/mcrf.2020011",
 url           = "https://www.aimsciences.org/article/doi/10.3934/mcrf.2020011",
 annote        = "We show that the joint spectral radius of a finite,
                  collection of nonnegative matrices can be bounded by the
                  eigenvalue of a non-linear operator. This eigenvalue
                  coincides with the ergodic constant of a risk-sensitive
                  control problem, or of an entropy game, in which the state
                  space consists of all switching sequences of a given
                  length. We show that, by increasing this length, we arrive
                  at a convergent approximation scheme to compute the joint
                  spectral radius. The complexity of this method is
                  exponential in the length of the switching sequences, but
                  it is quite insensitive to the size of the matrices,
                  allowing us to solve very large scale instances (several
                  matrices in dimensions of order 1000 within a minute). An
                  idea of this method is to replace a hierarchy of
                  optimization problems, introduced by Ahmadi, Jungers,
                  Parrilo and Roozbehani, by a hierarchy of nonlinear
                  eigenproblems. To solve the latter eigenproblems, we
                  introduce a projective version of Krasnoselskii--Mann
                  iteration. This method is of independent interest as it
                  applies more generally to the nonlinear eigenproblem for a
                  monotone positively homogeneous map. Here, this method
                  allows for scalability by avoiding the recourse to linear
                  or semidefinite programming techniques.",
}

@INPROCEEDINGS{GCM:CDC13,
 author        = "Gaye, Moussa and Chitour, Yacine and Mason, Paolo",
 authauthor    = "Yacine Chitour and Moussa Gaye and Paolo Mason",
 title         = "Properties of {B}arabanov norms and extremal trajectories
                  associated with continuous-time linear switched systems",
 booktitle     = "2013 IEEE 52nd Annual Conference on Decision and Control
                  (CDC)",
 organization  = "IEEE",
 year          = "2013",
 pages         = "716--721",
 hal_id        = "hal-00925301",
 hal_version   = "v1",
 doi           = "10.1109/CDC.2013.6759966",
 url           = "https://ieeexplore.ieee.org/document/6759966",
 annote        = "Consider continuous-time linear switched systems on
                  $\mathbb{R}^n$ associated with compact convex sets of
                  matrices. When the system is irreducible and the largest
                  Lyapunov exponent is equal to zero, a Barabanov norm always
                  exists. This paper deals with two sets of issues: (a)
                  properties of Barabanov norms such as uniqueness up to
                  homogeneity and strict convexity; (b) asymptotic behaviour
                  of the extremal solutions of the system. Regarding Issue
                  (a), we provide partial answers and propose two open
                  problems motivated by appropriate examples. As for Issue
                  (b), we establish, when $n = 3$, a Poincar{\'e}-Bendixson
                  theorem under a regularity assumption on the set of
                  matrices defining the system.",
}

@ARTICLE{Gelf:MatSb41:e,
 author        = "Gelfand, I.",
 title         = "Normierte {R}inge",
 journal       = "Rec. Math. [Mat. Sbornik] N. S.",
 year          = "1941",
 volume        = "9 (51)",
 pages         = "3--24",
 mrclass       = "46.3X",
 mrnumber      = "0004726 (3,51f)",
 mrreviewer    = "S. Bochner",
 zblnumber     = "67.0406.02",
 zblreviewer   = "K{\"o}the, G.; Prof. (Gie{\ss}en)",
 annote        = "Ein normierter Ring $R$ ist ein (bzgl. des K{\"o}rpers der
                  komplexen Zahlen) linearer, normierter vollst{\"a}ndiger
                  Raum im Sinne von Banach, in dem eine Multiplikation
                  erkl{\"a}rt ist, die ihn zu einem kommutativen Ring macht,
                  in dem ein Einheitselement $e$ vom Betrag 1 existiert und
                  stets $\| x\cdot y\|\leq \| x\| \cdot \| y\|$ gilt. Ist $R$
                  ein K{\"o}rper, so ist $R$ gleich dem K{\"o}rper der
                  komplexen Zahlen. Dies wird mit funktionentheoretischen
                  S{\"a}tzen bewiesen. Eine Funktion $x(\lambda )$, $\lambda
                  $ komplex, $x\in R$, hei{\ss}t analytisch in einem Bereich,
                  wenn dort $\displaystyle \lim_{h\to 0}\frac{x(\lambda
                  +h)-x(\lambda )}{h}$ im Sinn der Konvergenz nach der Norm
                  existiert. Die S{\"a}tze der gew{\"o}hnlichen
                  Funktionentheorie {\"u}bertragen sich auf diese Funktionen
                  leicht, da f{\"u}r jedes lineare Funktional $f(x)$ die
                  Funktion $f(x(\lambda ))$ analytisch im {\"u}blichen Sinn
                  wird. Jedes Ideal von $R$ ist in einem maximalen Ideal $M$
                  enthalten, dessen Restklassenring $R/M$ dem K{\"o}rper der
                  komplexen Zahlen isomorph ist. Jedem Element $x\in R$ wird
                  so die Funktion $x(M)$ zugeordnet, wobei $x(M)$ die
                  komplexe Zahl ist, die die Restklasse von $x$ in $R/M$
                  darstellt. Ist $x(M)$ nirgends Null, so besitzt $x$ ein
                  inverses Element. Die Funktion $x(M)$ ist dann und nur dann
                  identisch Null, wenn $x$ im Durchschnitt aller maximalen
                  Ideale von $R$ liegt. Notwendig und hinreichend daf{\"u}r
                  ist, da{\ss} $x$ ein verallgemeinertes nilpotentes Element
                  ist, f{\"u}r das $\root n\of{\|\,x^n\,\|}\to 0$ geht.\par
                  Die Menge $\mathfrak{M}$ aller maximalen Ideale von $R$
                  wird zu einem topologischen Raum, wenn man als eine
                  Umgebung $U(M_0)$ alle $M\in\mathfrak{M}$ mit
                  $|\,x_i(M)-x_i(M_0)\,|<\varepsilon $, $i = 1,\ldots, n,
                  x_1,\ldots, x_n$ aus $R$, einf{\"u}hrt. $\mathfrak{M}$ ist
                  bikompakt und gen{\"u}gt den Hausdorffschen Axiomen, die
                  $x(M)$ sind stetig auf $\mathfrak{M}$, also ist $R$
                  homomorph in den Ring $C(\mathfrak{M})$ aller auf
                  $\mathfrak{M}$ stetigen Funktionen abgebildet. Diese
                  Abbildung ist dann und nur dann eine Isomorphie, wenn $R$
                  kein Radikal, d. h. keine verallgemeinerten nilpotenten
                  Elemente, enth{\"a}lt. Ist $R$ separabel, so ist
                  $\mathfrak{M}$ metrisierbar. Besitzt $R$ eine endliche
                  Anzahl von Erzeugenden, so ist $\mathfrak{M}$ eine
                  abgeschlossene und beschr{\"a}nkte Teilmenge des
                  $n$-dimensionalen Raumes. Zwei normierte Ringe ohne
                  Radikal, die algebraisch isomorph sind, sind auch stetig
                  isomorph. $R$ ist dann und nur dann direkte Summe von
                  Idealen, die Ringe mit Einselementen sind, wenn
                  $\mathfrak{M}$ nicht zusammenh{\"a}ngend ist ($R$ mu{\ss}
                  dabei aber mit $x$ stets ein $y$ enthalten, dessen Funktion
                  $y(M)$ konjugiert komplex zu $x(M)$ ist). Untersuchungen
                  {\"u}ber analytische Funktionen von Elementen des Ringes
                  $R$.",
}

@ARTICLE{GPRSTT:P10,
 author        = "Gessesse, Hailegebriel and Popov, Alexey I. and Radjavi,
                  Heydar and Spinu, Eugeniu and Tcaciuc, Adi and Troitsky,
                  Vladimir G.",
 authauthor    = "Gessesse, Hailegebriel and Popov, Alexey and Radjavi, Heydar
                  and Spinu, Eugeniu and Tcaciuc, Adi and Troitsky,
                  Vladimir",
 title         = "Bounded indecomposable semigroups of non-negative matrices",
 journal       = "Positivity",
 fjournal      = "Positivity. An International Mathematics Journal Devoted to
                  Theory and Applications of Positivity",
 year          = "2010",
 volume        = "14",
 number        = "3",
 pages         = "383--394",
 issn          = "1385-1292",
 mrclass       = "20M20 (15B48 47B65 47D03)",
 mrnumber      = "2680502 (2011g:20103)",
 mrreviewer    = "Janusz Konieczny",
 zblnumber     = "1204.15041",
 zblreviewer   = "Mihail Voicu (Ia{\c{s}}i)",
 doi           = "10.1007/s11117-009-0024-5",
 url           = "https://link.springer.com/article/10.1007/s11117-009-0024-5",
 annote        = "Given a semigroup ${\mathfrak{S}}$ of non-negative, $n\times
                  n$, indecomposable matrices (i.e. for every pair $i,j \leq
                  n$ there exists ${S\in\mathfrak{S}}$ such that $S_{ij}\ne
                  0$), in this paper it is mainly shown that if there exists
                  a pair $k,l$ such that ${\{(S)_{kl} : S\in\mathfrak{S}\}}$
                  is bounded, then, after a simultaneous diagonal similarity,
                  all the entries are in $[0,1]$. Further, quantitative
                  versions of this result as well as extensions to
                  infinite-dimensional cases are provided, and finally, the
                  continuous case (convolution semigroup) is examined.",
}

@ARTICLE{GhaAna:TCS05,
 author        = "Gharavi, Reza and Anantharam, V.",
 title         = "An upper bound for the largest {L}yapunov exponent of a
                  {M}arkovian product of nonnegative matrices",
 journal       = "Theoret. Comput. Sci.",
 fjournal      = "Theoretical Computer Science",
 year          = "2005",
 volume        = "332",
 number        = "1-3",
 pages         = "543--557",
 coden         = "TCSDI",
 issn          = "0304-3975",
 mrclass       = "60J10 (15A45 15A48 68W15)",
 mrnumber      = "2122519 (2005i:60138)",
 zblnumber     = "1066.60066",
 zblreviewer   = "J{\'a}nos Acz{\'e}l (Waterloo/Ontario)",
 doi           = "10.1016/j.tcs.2004.12.025",
 url           = "https://www.sciencedirect.com/science/article/pii/S0304397504008205",
 annote        = "We derive an upper bound for the largest Lyapunov exponent
                  of a Markovian product of nonnegative matrices using
                  Markovian type counting arguments. The bound is expressed
                  as the maximum of a nonlinear concave function over a
                  finite-dimensional convex polytope of probability
                  distributions.",
}

@ARTICLE{Gil:UJMA19,
 author        = "Gil', Michael",
 title         = "A Bound for the Joint Spectral Radius of Operators in a
                  {H}ilbert Space",
 journal       = "Univers. J. Math. Appl.",
 fjournal      = "Universal Journal of Mathematics and Applications",
 address       = "Emrah Evren KARA",
 year          = "2019",
 volume        = "2",
 pages         = "94--99",
 issn          = "2619-9653",
 doi           = "10.32323/ujma.543952",
 url           = "https://dergipark.org.tr/ujma/issue/46498/543952",
 annote        = "We suggest a bound for the joint spectral radius of a finite
                  set of operators in a Hilbert space. In appropriate
                  situations that bound enables us to avoid complicated
                  calculations and gives a new explicit stability test for
                  the discrete time switched systems. The illustrative
                  example is given. Our results are new even in the finite
                  dimensional case.",
}

@PHDTHESIS{Gomes:2017,
 author        = "Gomes, Jo{\~a}o Tiago Assun{{\c{c}}{\~a}}o",
 authauthor    = "Gomes, Jo{\~a}o",
 title         = "Conjuntos maximizantes para sequ{\^e}ncias de
                  fun{{\c{c}}{\~o}}es e o {R}aio espectral conjunto",
 school        = "Universidade Estadual de Campinas, Instituto de
                  Matem{\'a}tica, Estat{\'{i}}stica e Computa{{\c{c}}{\~a}}o
                  Cient{\'{i}}fica",
 address       = "Campinas",
 year          = "2017",
 pagetotal     = "100",
 url           = "https://bdtd.ibict.br/vufind/Record/CAMP_01c3b1af1f91b867f24a22ac0650e6a3",
 annote        = "Em uma generaliza{{\c{c}}{\~a}}o da teoria de
                  otimiza{{\c{c}}{\~a}}o erg{\'o}dica para sequ{\^e}ncias de
                  fun{{\c{c}}{\~o}}es, desenvolvemos novas perspectivas no
                  que concerne {\`a} caracteriza{{\c{c}}{\~a}}o das
                  probabilidades maximizantes. Para isto obter, determinam-se
                  condi{{\c{c}}{\~o}}es suficientes que permitem destacar o
                  suporte de tais medidas por meio de vers{\~o}es
                  generalizadas do local de maximiza{{\c{c}}{\~a}}o e do
                  conjunto de Aubry. Al{\'e}m de aprimorar resultados
                  conhecidos na literatura da teoria de
                  otimiza{{\c{c}}{\~a}}o erg{\'o}dica sobre tais conjuntos
                  maximizantes, tamb{'{e}}m sugerimos extens{~{o}}es do
                  teorema de Atkinson e da no{{\c{c}}{\~a}}o de
                  suba{{\c{c}}{\~a}}o para o contexto das sequ{\^e}ncias de
                  fun{{\c{c}}{\~o}}es. Ao final, estudamos
                  condi{{\c{c}}{\~o}}es necess{\'a}rias e suficientes que
                  fornecem norma extremal de politopo e uma nova
                  formula{{\c{c}}{\~a}}o para a propriedade da finitude do
                  raio espectral conjunto.\par In a generalized version of
                  ergodic optimization theory for sequences of functions, we
                  develop new perspectives concerning the characterization of
                  maximizing probabilities. For this purpose, we provide
                  sufficient conditions that allow us to detach the support
                  of these measures by means of generalizations of the
                  maximizing locus and the Aubry set. Besides the fact that
                  we improve known results in the ergodic optimization theory
                  literature on these maximizing sets, we also suggest
                  extensions to Atkinson's theorem and to the notion of
                  sub-action for the context of sequences of functions. At
                  the end, we study necessary and sufficient conditions which
                  provide an extremal polytope norm and a new formulation for
                  the finiteness property of the joint spectral radius.",
}

@ARTICLE{Grip:LAA96,
 author        = "Gripenberg, Gustaf",
 title         = "Computing the joint spectral radius",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "1996",
 volume        = "234",
 pages         = "43--60",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15A60",
 mrnumber      = "1368770 (97c:15043)",
 mrreviewer    = "Zden{\v{e}}k Dost{\'a}l",
 zblnumber     = "0863.65017",
 zblreviewer   = "I. N. Molchanov (Kiev)",
 doi           = "10.1016/0024-3795(94)00082-4",
 url           = "https://www.sciencedirect.com/science/article/pii/0024379594000824",
 annote        = "This paper presents algorithms for finding an arbitrarily
                  small interval that contains the joint spectral radius of a
                  finite set of matrices. It also presents a numerical
                  criterion for verifying in certain cases that the joint
                  spectral radius is the maximum of the spectral radii of the
                  given matrices. Error bounds are derived for the case where
                  calculations are done with finite precision and the
                  matrices are not known exactly. The algorithms are
                  implemented and applied to estimate H{\"o}lder exponents of
                  the orthonormal wavelets $_{N}\phi$ constructed by
                  Daubechies for $3\le N\le 8$.",
}

@ARTICLE{GWZ:SIAMJMA05,
 author        = "Guglielmi, N. and Wirth, F. and Zennaro, M.",
 title         = "Complex polytope extremality results for families of
                  matrices",
 journal       = "SIAM J. Matrix Anal. Appl.",
 fjournal      = "SIAM Journal on Matrix Analysis and Applications",
 year          = "2005",
 volume        = "27",
 number        = "3",
 pages         = "721--743 (electronic)",
 issn          = "0895-4798",
 mrclass       = "93D09 (15A60 34D08)",
 mrnumber      = "2208331 (2007b:93106)",
 mrreviewer    = "M. M. Konstantinov",
 zblnumber     = "1099.15023",
 zblreviewer   = "Alexey Alimov (Moskva)",
 doi           = "10.1137/040606818",
 url           = "https://epubs.siam.org/doi/10.1137/040606818",
 annote        = "For finite families of complex $n\times n$ matrices, the
                  focus is paid on those families that satisfy the so-called
                  fitness conjecture, which was recently disproved in its
                  more general formulation. The authors conjecture that the
                  validity of the fitness conjecture for a finite family of
                  nondetective type is equivalent to the existence of an
                  extremal norm in the class of convex polytope norms.
                  However, they are not able to prove the small complex
                  polytope extremality theorem under some more restrictive
                  hypotheses on the underlying family of matrices.\par In
                  addition, the obtained results assure a certain fitness
                  property on the number of vertices of the unit ball of the
                  extremal complex polytope norm, which could be very useful
                  for the construction of suitable algorithms aimed at the
                  actual computation of the spectral radius of the
                  family.\par Even though the obtained results are mostly
                  theoretical in nature, they have potential impact in
                  applications. This potential lies in the fact that, if
                  there is a prior knowledge that a certain set $\mathcal{F}$
                  has an extremal polytope norm, then one could devise
                  algorithms for the computation of the spectral radius
                  $\rho(\mathcal{F})$ that rely on the computation of the
                  extremal points of the unit ball of the norm.",
}

@ARTICLE{GugZen:LAA01,
 author        = "Guglielmi, N. and Zennaro, M.",
 title         = "On the asymptotic properties of a family of matrices",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2001",
 volume        = "322",
 number        = "1-3",
 pages         = "169--192",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15A60 (65F35)",
 mrnumber      = "1804119 (2002b:15022)",
 mrreviewer    = "Rafikul Alam",
 zblnumber     = "0971.15016",
 zblreviewer   = "Rodica Covaci (Cluj-Napoca)",
 doi           = "10.1016/S0024-3795(00)00228-7",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379500002287",
 annote        = "The authors consider bounded families $\mathcal{F}$ of
                  complex $n \times n$-matrices. The concept of ``asymptotic
                  order'' is introduced and how the norm of products of
                  matrices behaves as the number of factors goes to infinity
                  is investigated. In the case of ``defective'' families
                  $\mathcal{F}$, using the asymptotic order one gets a deeper
                  knowledge of the asymptotic behaviour than by just
                  considering the so-called ``generalized spectral radius''.
                  With reference to the well-known finiteness conjecture for
                  finite families, the paper introduces the concepts of
                  ``spectrum-maximizing product'' and ``limit
                  spectrum-maximizing product'', showing that, for finite
                  families of $2\times 2$-matrices, defectivity is equivalent
                  to the existence of defective such limit products.",
}

@ARTICLE{GugZen:LAA03,
 author        = "Guglielmi, N. and Zennaro, M.",
 title         = "On the limit products of a family of matrices",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2003",
 volume        = "362",
 pages         = "11--27",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15A18",
 mrnumber      = "1955451 (2004b:15020)",
 zblnumber     = "1025.15040",
 zblreviewer   = "Alexey Alimov (Moskau)",
 doi           = "10.1016/S0024-3795(02)00460-3",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379502004603",
 annote        = "The authors~\cite{GugZen:LAA01} introduced the concept of
                  the limit spectrum-maximizing product (l.s.m.p.) for a
                  bounded family of complex $n\times n$ matrices and
                  characterized the defectivity of finite families of
                  $2\times 2$ matrices by the existence of defective l.s.m.\
                  products. At the end of [loc. cit.] it was conjectured that
                  this characterization should hold in general. \par Here the
                  authors prove that all nondefective bounded normalized
                  families of matrices always have an l.s.m.p.\ and show that
                  finite defective families of $3\times 3$ matrices always
                  have an spectrum-maximizing product (s.m.p.), but may not
                  have a defective l.s.m.p. Thus, the conjecture occured to
                  be false. Also, the authors construct a finite defective
                  family of $4\times 4$ matrices which does not have even any
                  l.s.m.p., regardless its possible defectivity.",
}

@ARTICLE{GugZen:SIAMJMAA09,
 author        = "Guglielmi, N. and Zennaro, M.",
 title         = "Finding extremal complex polytope norms for families of real
                  matrices",
 journal       = "SIAM J. Matrix Anal. Appl.",
 fjournal      = "SIAM Journal on Matrix Analysis and Applications",
 year          = "2009",
 volume        = "31",
 number        = "2",
 pages         = "602--620",
 coden         = "SJMAEL",
 issn          = "0895-4798",
 mrclass       = "15A60 (34D08 93D09)",
 mrnumber      = "2530266 (2010i:15053)",
 zblnumber     = "1197.93129",
 zblreviewer   = "Alexey Alimov (Moskva)",
 doi           = "10.1137/080715718",
 url           = "https://epubs.siam.org/doi/10.1137/080715718",
 annote        = "The paper considers finite families $\mathcal{F}$ of real
                  $n\times n$ matrices and, in particular, is focused on the
                  computation of the joint spectral radius
                  $\rho(\mathcal{F})$ through detection of an extremal norm
                  in the class of complex polytope norms whose unit balls are
                  balanced complex polytopes with a finite essential system
                  of vertices. Such a finiteness property is very useful in
                  view of the construction of efficient computational
                  algorithms. More precisely, the paper improves the results
                  obtained in a previous paper by the
                  authors~\cite{GWZ:SIAMJMA05}, which gave some conditions on
                  the family $\mathcal{F}$ to guarantee the existence of an
                  extremal complex polytope norm. Unfortunately, they exclude
                  unnecessarily many interesting cases of real families.
                  Therefore, here the conditions given in the previous paper
                  in question are relaxed in order to provide a~more
                  satisfactory treatment of the real case. A~modification of
                  the algorithm~\cite{GugZen:LAA08} for finding extremal
                  polytope norms of matrix families is put forward, its
                  computational relevance is discussed and illustrative
                  numerical examples are given.",
}

@ARTICLE{GugZen:LAA12,
 author        = "Guglielmi, N. and Zennaro, M.",
 title         = "On the asymptotic regularity of a family of matrices",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2012",
 volume        = "436",
 number        = "7",
 pages         = "2093--2104",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15A60",
 mrnumber      = "2889978",
 mrreviewer    = "Huiqiu Lin",
 zblnumber     = "1247.15006",
 zblreviewer   = "Wai-Shun Cheung (Hong Kong)",
 doi           = "10.1016/j.laa.2011.11.017",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379511007609",
 annote        = "In the paper bounded families $\mathcal{F}$ of complex
                  $n\times n$ matrices are considered. Sufficient conditions
                  are given under which the sequence
                  $\{\bar{\rho}_{k}(\mathcal{F})^{1/k}\}_{k\ge1}$, where
                  $\bar{\rho}_{k}(\mathcal{F})$ is the supremum of the
                  spectral radii of all possible products of $k$ matrices
                  chosen in $\mathcal{F}$, is convergent to its supremum
                  $\rho(\mathcal{F})$, the so-called (generalized) spectral
                  radius of $\mathcal{F}$. It is also illustrated a possible
                  practical application.\par Let $\mathcal{F}$ be a bounded
                  family of $n\times n$ complex matrices and
                  $\Sigma_{k}(\mathcal{F})=\{A_1 \cdots A_k : A_i
                  \in\mathcal{F}\}$. For each $k\ge 1$, define the number
                  $\bar{\rho}_{k}(\mathcal{F})=\sup_{Q\in\Sigma_{k}(\mathcal{F})}
                  \rho(Q)$ where $\rho(\cdot)$ denotes the spectral radius of
                  a matrix. The generalized spectral radius of $\mathcal{F}$
                  is $\rho(\mathcal{F})=\limsup_{k\to\infty}
                  \bar{\rho}_{k}(\mathcal{F})^{1/k}$. $\mathcal{F}$ is said
                  to be asymptotically regular if
                  \[\rho(\mathcal{F})=\lim_{k\to\infty}
                  \bar{\rho}_{k}(\mathcal{F})^{1/k}.\] $\mathcal{F}$ is said
                  to have finiteness property if there exists $k^{*} \ge 1$
                  and a $P\in\Sigma_{k^{*}}(\mathcal{F})$ such that
                  \[\rho(\mathcal{F})=\bar{\rho}_{k^{*}}(\mathcal{F})^{1/k^{*}}
                  =\rho(P)^{1/k^{*}}\] and the special matrix $P$ is called a
                  spectrum-maximizing product for $\mathcal{F}$. Theorem 3.1
                  states a sufficient condition on a matrix $P$ and $k^{*}>1$
                  such that
                  $\lim_{k\to\infty}\bar{\rho}_{k}(\mathcal{F})^{1/k} \ge
                  \rho(P)^{1/k^{*}}$. As a consequence, Corollary 3.1 states
                  a sufficient condition on a $\mathcal{F}$ with finiteness
                  property to be asymptotically regular. At the end of the
                  article, there is an application Theorem 5.2 which states
                  that if $\mathcal{F}$ is a bounded family of nonnegative
                  matrices with finiteness property and there exists a
                  primitive spectrum-maximizing product, then $\mathcal{F}$
                  is asymptotically regular.",
}

@ARTICLE{GugZen:JMAA15,
 author        = "Guglielmi, N. and Zennaro, M.",
 title         = "Canonical construction of polytope {B}arabanov norms and
                  antinorms for sets of matrices",
 journal       = "SIAM J. Matrix Anal. Appl.",
 fjournal      = "SIAM Journal on Matrix Analysis and Applications",
 year          = "2015",
 volume        = "36",
 number        = "2",
 pages         = "634--655",
 issn          = "0895-4798",
 mrclass       = "15A60 (15A18 90C90)",
 mrnumber      = "3352607",
 zblnumber     = "06444038",
 doi           = "10.1137/140962814",
 url           = "https://epubs.siam.org/doi/10.1137/140962814",
 annote        = "Barabanov norms have been introduced by
                  \emph{N.\,E.~Barabanov}~\cite{Bar:AIT88-2:e} and constitute
                  an important instrument in analyzing the joint spectral
                  radius of a family of matrices and related issues. However,
                  although they have been studied extensively, even in very
                  simple cases it is very difficult to construct them
                  explicitly (see, e.g.,
                  \emph{V.\,S.~Kozyakin}~\cite{Koz:DCDSB10}). In this paper
                  we give a canonical procedure to construct them exactly,
                  which associates a polytope extremal norm --- constructed
                  by using the methodologies described by
                  \emph{N.~Guglielmi}, \emph{F.~Wirth}, and
                  \emph{M.~Zennaro}~\cite{GWZ:SIAMJMA05} and
                  \emph{N.~Guglielmi} and
                  \emph{V.~Protasov}~\cite{GugProt:FCM13} --- to a polytope
                  Barabanov norm. Hence, the existence of a polytope
                  Barabanov norm has the same genericity of an extremal
                  polytope norm. Moreover, we extend the result to polytope
                  antinorms, which have been recently introduced to compute
                  the lower spectral radius of a finite family of matrices
                  having an invariant cone.",
}

@ARTICLE{GLP:FCM15,
 author        = "Guglielmi, Nicola and Laglia, Linda and Protasov, Vladimir",
 title         = "Polytope {L}yapunov functions for stable and for
                  stabilizable {LSS}",
 journal       = "Found. Comput. Math.",
 fjournal      = "Foundations of Computational Mathematics. The Journal of the
                  Society for the Foundations of Computational Mathematics",
 year          = "2017",
 volume        = "17",
 number        = "2",
 pages         = "567--623",
 eprinttype    = "arXiv",
 eprint        = "1406.5927",
 issn          = "1615-3375",
 mrclass       = "93D30 (15A60 34D08 90C90)",
 mrnumber      = "3627457",
 mrreviewer    = "Kaiqi Xiong",
 zblnumber     = "1361.93050",
 doi           = "10.1007/s10208-015-9301-9",
 url           = "https://link.springer.com/article/10.1007/s10208-015-9301-9",
 annote        = "We present a new approach for constructing polytope Lyapunov
                  functions for continuous-time linear switching systems
                  (LSS). This allows us to decide the stability of LSS and to
                  compute the Lyapunov exponent with a good precision in
                  relatively high dimensions. The same technique is also
                  extended for stabilizability of positive systems by
                  evaluating a polytope concave Lyapunov function
                  (``antinorm'') in the cone. The method is based on a
                  suitable discretization of the underlying continuous system
                  and provides both a lower and an upper bound for the
                  Lyapunov exponent. The absolute error in the Lyapunov
                  exponent computation (the distance between lower and upper
                  bound) is estimated from above and proved to be linear in
                  the dwell time. The practical efficiency of the new method
                  is demonstrated in several examples and in the list of
                  numerical experiments with randomly generated matrices of
                  dimensions up to $10$ (for general linear systems) and up
                  to $100$ (for positive systems). The development of the
                  method is based on several theoretical results proved in
                  the paper: the existence of monotone invariant norms and
                  antinorms for positively irreducible systems, the
                  equivalence of all contractive norms for stable systems and
                  the linear convergence theorem.",
}

@ARTICLE{GMW:LAA17,
 author        = "Guglielmi, Nicola and Mason, Oliver and Wirth, Fabian",
 title         = "Barabanov norms, {L}ipschitz continuity and monotonicity for
                  the max algebraic joint spectral radius",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2018",
 volume        = "550",
 pages         = "37--58",
 eprinttype    = "arXiv",
 eprint        = "1705.02008",
 issn          = "0024-3795",
 mrclass       = "15A60 (15B48)",
 mrnumber      = "3786246",
 mrreviewer    = "Lei Cao",
 zblnumber     = "1390.15112",
 doi           = "10.1016/j.laa.2018.01.042",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379518301526",
 annote        = "We present several results describing the interplay between
                  the max algebraic joint spectral radius (JSR) for compact
                  sets of matrices and suitably defined matrix norms. In
                  particular, we extend a classical result for the
                  conventional algebra, showing that the JSR can be described
                  in terms of induced norms of the matrices in the set. We
                  also show that for a set generating an irreducible
                  semigroup (in a cone-theoretic sense), a monotone Barabanov
                  norm always exists. This fact is then used to show that the
                  max algebraic JSR is locally Lipschitz continuous on the
                  space of compact irreducible sets of matrices with respect
                  to the Hausdorff distance. We then prove that the JSR is
                  Hoelder continuous on the space of compact sets of
                  nonnegative matrices. Finally, we prove a strict
                  monotonicity property for the max algebraic JSR that echoes
                  a fact for the classical JSR.",
}

@ARTICLE{GugProt:FCM13,
 author        = "Guglielmi, Nicola and Protasov, Vladimir",
 title         = "Exact computation of joint spectral characteristics of
                  linear operators",
 journal       = "Found. Comput. Math.",
 fjournal      = "Foundations of Computational Mathematics. The Journal of the
                  Society for the Foundations of Computational Mathematics",
 year          = "2013",
 volume        = "13",
 number        = "1",
 pages         = "37--97",
 eprinttype    = "arXiv",
 eprint        = "1106.3755",
 issn          = "1615-3375",
 mrclass       = "65F15 (15A60 47A10 90C90)",
 mrnumber      = "3009529",
 mrreviewer    = "Marius Ghergu",
 zblnumber     = "06153962",
 doi           = "10.1007/s10208-012-9121-0",
 url           = "https://link.springer.com/article/10.1007/s10208-012-9121-0",
 annote        = "We address the problem of the exact computation of two joint
                  spectral characteristics of a family of linear operators,
                  the joint spectral radius (JSR) and the lower spectral
                  radius (LSR), which are well-known different
                  generalizations to a set of operators of the usual spectral
                  radius of a linear operator. In this paper we develop a
                  method which -- under suitable assumptions -- allows us to
                  compute the JSR and the LSR of a finite family of matrices
                  exactly. We remark that so far no algorithm has been
                  available in the literature to compute the LSR exactly. The
                  paper presents necessary theoretical results on extremal
                  norms (and on extremal antinorms) of linear operators,
                  which constitute the basic tools of our procedures, and a
                  detailed description of the corresponding algorithms for
                  the computation of the JSR and LSR (the last one restricted
                  to families sharing an invariant cone). The algorithms are
                  easily implemented, and their descriptions are short. If
                  the algorithms terminate in finite time, then they
                  construct an extremal norm (in the JSR case) or antinorm
                  (in the LSR case) and find their exact values; otherwise,
                  they provide upper and lower bounds that both converge to
                  the exact values. A theoretical criterion for termination
                  in finite time is also derived. According to numerical
                  experiments, the algorithm for the JSR finds the exact
                  value for the vast majority of matrix families in
                  dimensions $\leq$20. For nonnegative matrices it works
                  faster and finds the JSR in dimensions of order 100 within
                  a few iterations; the same is observed for the algorithm
                  computing the LSR. To illustrate the efficiency of the new
                  method, we apply it to give answers to several conjectures
                  which have been recently stated in combinatorics, number
                  theory, and formal language theory.",
}

@INPROCEEDINGS{GugProt:AIP16,
 author        = "Guglielmi, Nicola and Protasov, Vladimir",
 title         = "Computing {L}yapunov exponents of switching systems",
 booktitle     = "AIP Conference Proceedings {\textbf{1738}}, 020006",
 publisher     = "AIP Publishing",
 address       = "Rhodes, Greece",
 year          = "2016",
 pages         = "020006--1--020006--4",
 doi           = "10.1063/1.4951750",
 url           = "https://aip.scitation.org/doi/abs/10.1063/1.4951750",
 annote        = "We discuss a new approach for constructing polytope Lyapunov
                  functions for continuous-time linear switching systems. The
                  method we propose allows to decide the uniform stability of
                  a switching system and to compute the Lyapunov exponent
                  with an arbitrary precision. The method relies on the
                  discretization of the system and provides --- for any given
                  discretization stepsize --- a lower and an upper bound for
                  the Lyapunov exponent. The efficiency of the new method is
                  illustrated by numerical examples. For a more extensive
                  discussion we remand the reader to~\cite{GWZ:SIAMJMA05}.",
}

@ARTICLE{GugProt:MC23,
 author        = "Nicola Guglielmi and Vladimir Protasov",
 title         = "Computing the spectral gap of a family of matrices",
 journal       = "Math. Comp.",
 fjournal      = "Mathematics of Computation",
 volume        = "93",
 year          = "2023",
 number        = "345",
 pages         = "259--291",
 issn          = "0025-5718,1088-6842",
 mrclass       = "65F45 (15A42 15A60 47D03 93C30 93D20 93D50)",
 mrnumber      = "4654622",
 zblnumber     = "07753429",
 doi           = "10.1090/mcom/3856",
 url           = "https://doi.org/10.1090/mcom/3856",
 annote        = "For a single matrix (operator) it is well-known that the
                  spectral gap is an important quantity, as well as its
                  estimate and computation. Here we consider, for the first
                  time in the literature, the computation of its extension to
                  a finite family of matrices, in other words the difference
                  between the joint spectral radius (in short JSR, which we
                  call here the first Lyapunov exponent) and the second
                  Lyapunov exponent (denoted as SLE). The knowledge of joint
                  spectral characteristics and of the spectral gap of a
                  family of matrices is important in several applications, as
                  in the analysis of the regularity of wavelets,
                  multiplicative matrix semigroups and the convergence speed
                  in consensus algorithms. As far as we know the methods we
                  propose are the first able to compute this quantity to any
                  given accuracy.\par For computation of the spectral gap one
                  needs first to compute the JSR. A popular tool that is used
                  to this purpose is the invariant polytope algorithm, which
                  relies on the finiteness property of the family of
                  matrices, when this holds true.\par In this paper we show
                  that the SLE may not possess the finiteness property,
                  although it can be efficiently approximated with an
                  arbitrary precision. The corresponding algorithm and two
                  effective estimates are presented. Moreover, we prove that
                  the SLE possesses a weak finiteness property, whenever the
                  leading eigenvalue of the dominant product is real. This
                  allows us to find in certain situations the precise value
                  of the SLE. Numerical results are demonstrated along with
                  applications in the theory of multiplicative matrix
                  semigroups and in the wavelets theory.",
}

@ARTICLE{GugProt:SIAMJMAA16,
 author        = "Guglielmi, Nicola and Protasov, Vladimir {\relax{}Yu}.",
 authauthor    = "Guglielmi, Nicola and Protasov, Vladimir",
 title         = "Invariant Polytopes of Sets of Matrices with Application to
                  Regularity of Wavelets and Subdivisions",
 journal       = "SIAM J. Matrix Anal. Appl.",
 fjournal      = "SIAM Journal on Matrix Analysis and Applications",
 year          = "2016",
 volume        = "37",
 number        = "1",
 pages         = "18--52",
 eprinttype    = "arXiv",
 eprint        = "1502.01192",
 issn          = "0895-4798",
 mrclass       = "15A60 (15A18 90C90)",
 mrnumber      = "3447127",
 zblnumber     = "06548831",
 doi           = "10.1137/15M1006945",
 url           = "https://epubs.siam.org/doi/10.1137/15M1006945",
 annote        = "We generalize the recent invariant polytope algorithm for
                  computing the joint spectral radius and extend it to a
                  wider class of matrix sets. This, in particular, makes the
                  algorithm applicable to sets of matrices that have finitely
                  many spectrum maximizing products. A criterion of
                  convergence of the algorithm is proved.\par As an
                  application we solve two challenging computational open
                  problems. First we find the regularity of the Butterfly
                  subdivision scheme for various parameters $\omega$. In the
                  ``most regular'' case $\omega = \frac{1}{16}$, we prove
                  that the limit function has H{\"o}lder exponent $2$ and its
                  derivative is ``almost Lipschitz'' with logarithmic
                  factor~$2$. Second we compute the H{\"o}lder exponent of
                  Daubechies wavelets of high order.",
}

@INPROCEEDINGS{GugZen:CDC05,
 author        = "Guglielmi, Nicola and Zennaro, Marino",
 title         = "Polytope Norms and Related Algorithms for the Computation of
                  the Joint Spectral Radius",
 booktitle     = "Proceedings of the 44th {IEEE} Conference on Decision and
                  Control and European Control Conference 2005, Seville,
                  Spain, December 12--15",
 year          = "2005",
 pages         = "3007--3012",
 doi           = "10.1109/CDC.2005.1582622",
 annote        = "We address the problem of the computation of the spectral
                  radius of a family of matrices. We briefly describe the
                  extension of the concept of polytope to the complex space
                  and outline the main geometric properties of such an
                  object. Then we consider the norms determined by the
                  complex polytopes and illustrate a possible algorithm for
                  the approximation of the joint spectral radius of a family
                  of matrices which is based on these complex polytope norms.
                  As an example for our technique we consider the set of two
                  matrices recently analyzed by Blondel, Nesterov and Theys
                  to disprove the finiteness conjecture.",
}

@ARTICLE{GugZen:JCA07,
 author        = "Guglielmi, Nicola and Zennaro, Marino",
 title         = "Balanced complex polytopes and related vector and matrix
                  norms",
 journal       = "J. Convex Anal.",
 fjournal      = "Journal of Convex Analysis",
 year          = "2007",
 volume        = "14",
 number        = "4",
 pages         = "729--766",
 issn          = "0944-6532",
 mrclass       = "52B99",
 mrnumber      = "2350813",
 mrreviewer    = "Keith Ball",
 zblnumber     = "1128.52010",
 url           = "https://www.heldermann.de/JCA/JCA14/JCA144/jca14043.htm",
 annote        = "In this paper we study the notion of \emph{balanced complex
                  polytope} as a generalization of a symmetric real polytope
                  to the complex space $\mathbb{C}^n$. We pay particular
                  attention to the geometric properties of such complex
                  polytopes and of their counterparts in the \emph{adjoint}
                  form. In particular, we stress the differences occurring
                  with respect to the well-known real case. We also introduce
                  and discuss the related definitions of \emph{complex
                  polytope norm} and \emph{adjoint complex polytope norm}.",
}

@ARTICLE{GugZen:LAA08,
 author        = "Guglielmi, Nicola and Zennaro, Marino",
 title         = "An algorithm for finding extremal polytope norms of matrix
                  families",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2008",
 volume        = "428",
 number        = "10",
 pages         = "2265--2282",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15A60 (52A07)",
 mrnumber      = "2405244 (2009b:15071)",
 mrreviewer    = "Edward S. Becerra",
 zblnumber     = "1139.65027",
 doi           = "10.1016/j.laa.2007.07.009",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379507003126",
 annote        = "The problem of the computation of the joint spectral radius
                  of a finite set of matrices is considered. We present an
                  algorithm which, under some suitable assumptions, is able
                  to check if a certain product in the multiplicative
                  semigroup is spectrum maximizing. The algorithm proceeds by
                  attempting to construct a suitable extremal norm for the
                  family, namely a complex polytope norm.\par As examples for
                  testing our technique, we first consider the set of two
                  $2$-dimensional matrices recently analyzed by
                  \emph{V.\,D.~Blondel}, \emph{J.~Theys} and
                  \emph{A.\,A.~Vladimirov}~\cite{BTV:SIAMJMA03} to disprove
                  the finiteness conjecture, and then a set of
                  $3$-dimensional matrices arising in the zero-stability
                  analysis of the $4$-step backward differentiation formula
                  for ordinary differential equations.",
}

@INCOLLECTION{GugZen:LNM14,
 author        = "Guglielmi, Nicola and Zennaro, Marino",
 title         = "Stability of Linear Problems: {J}oint Spectral Radius of
                  Sets of Matrices",
 booktitle     = "Current Challenges in Stability Issues for Numerical
                  Differential Equations",
 series        = "Lecture Notes in Mathematics",
 publisher     = "Springer",
 year          = "2014",
 pages         = "265--313",
 mrclass       = "65F15 (15A60 39B82)",
 mrnumber      = "3204994",
 zblnumber     = "1318.65081",
 doi           = "10.1007/978-3-319-01300-8_5",
 url           = "https://link.springer.com/chapter/10.1007/978-3-319-01300-8_5",
 annote        = "It is well known that the stability analysis of step-by-step
                  numerical methods for differential equations often reduces
                  to the analysis of linear difference equations with
                  variable coefficients. This class of difference equations
                  leads to a family $\mathcal{F}$ of matrices depending on
                  some parameters and the behaviour of the solutions depends
                  on the convergence properties of the products of the
                  matrices of $\mathcal{F}$. To date, the techniques mainly
                  used in the literature are confined to the search for a
                  suitable norm and for conditions on the parameters such
                  that the matrices of $\mathcal{F}$ are contractive in that
                  norm. In general, the resulting conditions are more
                  restrictive than necessary. An alternative and more
                  effective approach is based on the concept of joint
                  spectral radius of the family $\mathcal{F}$,
                  $\rho(\mathcal{F})$. It is known that all the products of
                  matrices of $\mathcal{F}$ asymptotically vanish if and only
                  if $\rho(\mathcal{F})<1$. The aim of this chapter is that
                  to discuss the main theoretical and computational aspects
                  involved in the analysis of the joint spectral radius and
                  in applying this tool to the stability analysis of the
                  discretizations of differential equations as well as to
                  other stability problems. In particular, in the last
                  section, we present some recent heuristic techniques for
                  the search of optimal products in finite families, which
                  constitutes a fundamental step in the algorithms which we
                  discuss. The material we present in the final section is
                  part of an original research which is in progress and is
                  still unpublished.",
}

@ARTICLE{GugZen:LAA20,
 author        = "Guglielmi, Nicola and Zennaro, Marino",
 title         = "An antinorm theory for sets of matrices: {B}ounds and
                  approximations to the lower spectral radius",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2020",
 volume        = "607",
 pages         = "89--117",
 issn          = "0024-3795",
 mrclass       = "15A60 (15A18 52A20 52A41)",
 mrnumber      = "4137719",
 zblnumber     = "07309723",
 doi           = "10.1016/j.laa.2020.07.037",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379520303761",
 annote        = "For the computation of the lower spectral radius of a finite
                  family of matrices that shares an invariant cone, two
                  recent papers by Guglielmi and
                  Protasov~\cite{GugProt:FCM13} and Guglielmi and
                  Zennaro~\cite{GugZen:JMAA15} make use of so-called
                  antinorms. Antinorms are continuous, nonnegative,
                  positively homogeneous and superadditive functions defined
                  on the cone and turn out to be related to the lower
                  spectral radius of the family in a similar way as norms are
                  related to the joint spectral radius. In this paper, we
                  revisit the theory of antinorms in a systematic way,
                  filling in some theoretical holes, correcting a common
                  mistake present in the literature and adding some new
                  properties and results. In particular, we prove that, under
                  suitable assumptions, the lower spectral radius is
                  characterized by a Gelfand type limit computed on an
                  antinorm.",
}

@ARTICLE{Guinand:PAMS82,
 author        = "Guinand, P. S.",
 title         = "On quasinilpotent semigroups of operators",
 journal       = "Proc. Amer. Math. Soc.",
 fjournal      = "Proceedings of the American Mathematical Society",
 year          = "1982",
 volume        = "86",
 number        = "3",
 pages         = "485--486",
 coden         = "PAMYAR",
 issn          = "0002-9939",
 mrclass       = "47D05",
 mrnumber      = "671220",
 mrreviewer    = "Eric Nordgren",
 zblnumber     = "0503.47038",
 doi           = "10.2307/2044453",
 url           = "https://www.ams.org/journals/proc/1982-086-03/S0002-9939-1982-0671220-5/home.html",
 annote        = "A pair of operators $A, B$ is constructed such that the
                  semigroup generated by them consists of operators which are
                  nilpotent of index $3$. The sum $A+B$, however, is not
                  quasinilpotent.\par{}As noted in \cite{ShulTur:SM02} this
                  implies that the joint spectral radius of the set $\{A,B\}$
                  is strictly greater than zero, while the generalized
                  spectral radius of the same set of operators is equal to
                  zero. Thus, in this case the Berger--Wang theorem
                  \cite{BerWang:LAA92} is not valid.",
}

@ARTICLE{GurMas:LAA11,
 author        = "Gursoy, Buket Benek and Mason, Oliver",
 authauthor    = "Gursoy, Buket and Mason, Oliver",
 title         = "{$P^1_{\max}$} and {$S_{\max}$} properties and asymptotic
                  stability in the max algebra",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2011",
 volume        = "435",
 number        = "5",
 pages         = "1008--1018",
 issn          = "0024-3795",
 mrclass       = "15A80 (39A30)",
 mrnumber      = "2807214",
 mrreviewer    = "Minghua Lin",
 zblnumber     = "1223.15034",
 zblreviewer   = "Alexander Guterman",
 doi           = "10.1016/j.laa.2011.02.054",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379511001893?via\%3Dihub",
 annote        = "A matrix $A\in\mathbb{R}^{n\times n}_{+}$ is said to be a
                  $P^1_{\max}$-matrix if
                  $\mathop{\mathrm{per}}_{\max}(B)\le1$ for any principal
                  submatrix $B$ of $A$. Here \[
                  \mathop{\mathrm{per}}\nolimits_{\max}(B)= \max_{\sigma\in
                  S_{n}}\otimes^{n}_{i=1} a_{i,\sigma(i)} \] is the
                  max-algebraic permanent function. The authors define the
                  class of $P^1_{\max}$-matrices and obtain its
                  characterization. Then using the approach of \emph{Y.~Song,
                  M.\,S.~Gowda, G.~Ravindran} (On some properties of
                  $P$-matrix sets. Linear Algebra Appl., \textbf{290} (1999),
                  pp. 237--246) in the new max-algebraic context these
                  results are extended for the sets of matrices. To do this
                  the authors introduce row-$P^1_{\max}$-property and
                  $S_{\max}$-property by the analogy with the corresponding
                  definitions in the conventional arithmetics. It is proved
                  that $S_{\max}$ property for a set of matrices is related
                  to the stability of its max-convex hull. The above results
                  are then applied to the study of stability questions for
                  discrete-time systems and certain systems of difference
                  equations over max-algebras.",
}

@INPROCEEDINGS{GurSam:CDC05,
 author        = "Gurvits, L. and Samorodnitsky, A.",
 title         = "A note on common quadratic {L}yapunov functions for linear
                  inclusions: {E}xact results and Open Problems",
 booktitle     = "Proceedings of the 44th IEEE Conference on Decision and
                  Control, 2005 and 2005 European Control Conference.
                  CDC-ECC'05, 15-15 Dec.",
 publisher     = "IEEE",
 address       = "Seville, Spain, Spain",
 year          = "2005",
 pages         = "2350--2355",
 doi           = "10.1109/CDC.2005.1582513",
 url           = "https://ieeexplore.ieee.org/document/1582513",
 annote        = "We prove several exact results on approximability of joint
                  spectral radius by matrix norms induced by Euclidean norms.
                  We point out, perhaps for the first time in this context, a
                  difference between complex and real cases. New connections
                  of joint spectral radius to convex geometry and
                  combinatorics are established. Several open problems are
                  posed.",
}

@ARTICLE{GSM:IEEETAC07,
 author        = "Gurvits, L. and Shorten, R. and Mason, O.",
 title         = "On the stability of switched positive linear systems",
 journal       = "IEEE Trans. Automat. Control",
 fjournal      = "Institute of Electrical and Electronics Engineers.
                  Transactions on Automatic Control",
 year          = "2007",
 volume        = "52",
 number        = "6",
 pages         = "1099--1103",
 coden         = "IETAA9",
 issn          = "0018-9286",
 mrclass       = "34D20 (34A36 93D05)",
 mrnumber      = "2329904 (2008e:34121)",
 zblnumber     = "1366.93436",
 doi           = "10.1109/TAC.2007.899057",
 url           = "https://ieeexplore.ieee.org/document/4237306",
 annote        = "It was recently conjectured that the Hurwitz stability of,
                  the, convex hull of a set of Metzler matrices is a
                  necessary and sufficient condition for the asymptotic
                  stability of the associated switched linear system under
                  arbitrary switching. In this note, we show that (1) this
                  conjecture is true for systems constructed from a pair of
                  second-order Metzler matrices; (2) the conjecture is true
                  for systems constructed from an arbitrary finite number of
                  second-order Metzler matrices; and (3) the conjecture is in
                  general false for higher order systems. The implications of
                  our results, both for the design of switched positive
                  linear systems, and for research directions that arise as a
                  result of our work, are discussed toward the end of the
                  note.",
}

@ARTICLE{Gurv:LAA95,
 author        = "Gurvits, Leonid",
 title         = "Stability of discrete linear inclusion",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "1995",
 volume        = "231",
 pages         = "47--85",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "93D05 (15A60 93B25 93C55)",
 mrnumber      = "1361100 (96i:93056)",
 mrreviewer    = "Mihail Voicu",
 zblnumber     = "0845.68067",
 doi           = "10.1016/0024-3795(95)90006-3",
 url           = "https://www.sciencedirect.com/science/article/pii/0024379595900063",
 annote        = "Let $M = \{A_i\}$ be a set of linear operators on
                  $\mathbb{R}^{n}$. The discrete linear inclusion DLI$(M)$ is
                  the set of possible trajectories $(x_i: i\ge 0)$ such that
                  $x_n = A_{i_{n}}A_{i_{n-1}}\cdots A_{i_{1}}x_{0}$ where
                  $x_{0}\in\mathbb{R}^{n}$ and $A_{i_{j}}\in M$. We study
                  several notions of stability for DLI$(M)$, including
                  absolute asymptotic stability (AAS), which is that all
                  products $A_{i_{n}}\cdots A_{i_{1}}\to 0$ as $n\to\infty$.
                  We mainly study the case that $M$ is a finite set. We give
                  criteria for the various forms of stability. Two new
                  approaches are taken: one relates the question of AAS of
                  DLI$(M)$ to formal language theory and finite automata,
                  while the second connects the AAS property to the structure
                  of a Lie algebra associated to the elements of $M$. More
                  generally, the discrete linear inclusion DLI$(M)$ makes
                  sense for $M$ contained in a Banach algebra $\mathscr{B}$.
                  We prove some results for AAS in this case, and give
                  counterexamples showing that some results valid for finite
                  sets of operators on $\mathbb{R}^{n}$ are not true for
                  finite sets $M$ in a general Banach algebras
                  $\mathscr{B}$.\par In particular, an example is given
                  demonstrating that the Berger--Wang
                  formula~\cite{BerWang:LAA92} need not be true for finite
                  sets of linear operators on an infinite dimensional Hilbert
                  space.",
}

@ARTICLE{GurRod:SIAMJMAA97,
 author        = "Gurvits, Leonid and Rodman, Leiba",
 title         = "Convergence of polynomially bounded semigroups of matrices",
 journal       = "SIAM J. Matrix Anal. Appl.",
 fjournal      = "SIAM Journal on Matrix Analysis and Applications",
 year          = "1997",
 volume        = "18",
 number        = "2",
 pages         = "360--368",
 issn          = "0895-4798",
 mrclass       = "15A30 (15A04)",
 mrnumber      = "1437336",
 mrreviewer    = "Walter S. Sizer",
 zblnumber     = "0876.15013",
 zblreviewer   = "F. Uhlig",
 doi           = "10.1137/S089547989528939X",
 url           = "https://epubs.siam.org/doi/10.1137/S089547989528939X",
 annote        = "A set of $n\times n$ matrices is called pointwise convergent
                  if for each vector $x$ there is an infinite product $A_{w}$
                  of matrices from the set with $A_{w}x\to0$ as the length of
                  $w$ goes to infinity. It is called uniformly convergent if
                  the factors in $A_{w}$ do not depend on $x$. It is shown
                  that if the matrix set generates a polynomially bounded
                  semigroup with $\sup_{|w|=k}|A_{w}|_{2}\le Ck^{p}$, then
                  the two notions of convergence coincide. This result is
                  also proved for certain sets of nonlinear maps on
                  finite-dimensional real or complex vector spaces.",
}

@ARTICLE{HNRRR:HJM91,
 author        = "Hadwin, D. and Nordgren, E. and Radjabalipour, M. and
                  Radjavi, H. and Rosenthal, P.",
 title         = "On simultaneous triangularization of collections of
                  operators",
 journal       = "Houston J. Math.",
 fjournal      = "Houston Journal of Mathematics",
 year          = "1991",
 volume        = "17",
 number        = "4",
 pages         = "581--602",
 coden         = "HJMADZ",
 issn          = "0362-1588",
 mrclass       = "47A15 (47B99 47D25)",
 mrnumber      = "1147275 (93a:47004)",
 mrreviewer    = "Kenneth R. Davidson",
 zblnumber     = "0784.47032",
 zblreviewer   = "T. Nakazi (Sapporo)",
 url           = "https://www.math.uh.edu/~hjm/v017n4/0581HADWIN.pdf",
 annote        = "Let $G$ be a Banach algebra of operators and
                  $\mathop{\text{Rad}}G$ the Jacobson radical of $G$. The
                  authors show that if $G/\mathop{\text{Rad}}G$ is
                  commutative, weakly closed and contains a maximal Abelian
                  self-adjoint algebra, then it is triangularizable. They
                  also show that a direct integral of weakly closed algebras
                  is hypertriangularizable if and only if almost every of the
                  algebras is hypertriangularizable. The analogous result for
                  triangularizability is false. Theorems on the existence of
                  invariant subspaces yield theorems on triangularizability.
                  They show that the problem of existence of invariant
                  subspaces for algebras that contain a bilateral and an
                  injective backward bilateral weighted shift is related to
                  the periodicity of the weights of the weighted shift.",
}

@ARTICLE{Hajnal:MPCPS76,
 author        = "Hajnal, J.",
 title         = "On products of non-negative matrices",
 journal       = "Math. Proc. Camb. Philos. Soc.",
 fjournal      = "Mathematical Proceedings of the Cambridge Philosophical
                  Society",
 year          = "1976",
 volume        = "79",
 issue         = "03",
 pages         = "521--530",
 pagetotal     = "10",
 month         = may,
 issn          = "1469-8064",
 mrclass       = "15A48 (60J10)",
 mrnumber      = "396628",
 mrreviewer    = "E. Seneta",
 zblnumber     = "0333.15013",
 doi           = "10.1017/S030500410005252X",
 url           = "https://dx.doi.org/10.1017/S030500410005252X",
 annote        = "The main theme of this paper is that under wide conditions,
                  the product of a large number of square non-negative
                  matrices is close to a positive matrix of rank $1$. (A
                  non-negative matrix is one whose elements are non-negative
                  real numbers). The results may be viewed as an extension of
                  the Perron--Probenius theory for powers of non-negative
                  matrices. Previous results in this field are reviewed in
                  E.~Seneta's book ``Non-negative matrices'' (1973). The
                  present paper applies Birkhoff's theory of positive
                  matrices as operators possessing a contraction property.
                  The tendency of non-negative matrix products towards rank
                  $1$ is referred to as ergodicity (a term derived in this
                  context from the theory of non-homogeneous Markov chains)
                  and is here defined in such a manner as to apply to
                  sequences of matrix products in which successive terms are
                  formed by inserting an additional matrix in any place (and
                  not merely at the extreme right or left). --- The paper
                  presents results for two sorts of sequences of products
                  called ``ergodic sequences'' and ``ergodic sets''. An
                  ergodic sequence must contain a subsequence of matrices
                  having no zero elements. The paper proves such results as
                  the following. Suppose that (i) $A_{1},A_{2},\ldots$ are
                  non-negative matrices each of which has at least one
                  strictly positive element in each row and each column and
                  (ii) $\sum_{k}\frac{m_{k}}{M_{k}}=\infty$ (where $m_{k}$
                  and $M_{k}$ are the smallest and largest element of
                  $A_{k}$). Then if $H_{n}$ is a product of the first $n$ of
                  the matrices $A_{k}$ multiplied in any order, the sequence
                  $\{H_{n}\}$ is a product of the first $n$ of the matrices
                  $A_{k}$ multiplied in any order, the sequence $\{H_{n}\}$
                  is ergodic, i.e. will be arbitrarily dose to matrix of rank
                  $1$ for sufficiently large $n$. --- The theorems concerning
                  ``ergodic sets'' generalize results which have emerged in
                  some fields of application such as the weak ergodicity
                  theorem (Coale--Lopez) theorem) of demography or the Markov
                  systems in Paz'e work on probabilistic automata.",
}

@ARTICLE{HR:APM12,
 author        = "Hanna, Yousry S. and Ragheb, Samya F.",
 authauthor    = "Hanna, Yousry and Ragheb, Samya",
 title         = "On the Infinite Products of Matrices",
 journal       = "Advances in Pure Mathematics",
 year          = "2012",
 volume        = "5",
 number        = "2",
 pages         = "349--353",
 doi           = "10.4236/apm.2012.25050",
 url           = "https://www.scirp.org/journal/paperinformation.aspx?paperid=22805",
 annote        = "In different fields in space researches, scientists are in
                  need to deal with the product of matrices. In this paper,
                  we develop conditions under which a product
                  $\prod_{i=0}^{\infty}$ of matrices chosen from a possibly
                  infinite set of matrices $M=\{P_{j},~ j\in J\}$ converges.
                  There exists a vector norm such that all matrices in $M$
                  are no expansive with respect to this norm and also a
                  subsequence $\{i_{k}\}_{k=0}^{\infty}$ of the sequence of
                  nonnegative integers such that the corresponding sequence
                  of operators $\{P_{ik}\}_{k=0}^{\infty}$ converges to an
                  operator which is paracontracting with respect to this
                  norm. The continuity of the limit of the product of
                  matrices as a function of the sequences
                  $\{i_{k}\}_{k=0}^{\infty}$ is deduced. The results are
                  applied to the convergence of inner-outer iteration schemes
                  for solving singular consistent linear systems of
                  equations, where the outer splitting is regular and the
                  inner splitting is weak regular.",
}

@ARTICLE{HMST:AdvMath11,
 author        = "Hare, Kevin G. and Morris, Ian D. and Sidorov, Nikita and
                  Theys, Jacques",
 authauthor    = "Hare, Kevin and Morris, Ian and Sidorov, Nikita and Theys,
                  Jacques",
 title         = "An explicit counterexample to the {L}agarias-{W}ang
                  finiteness conjecture",
 journal       = "Adv. Math.",
 fjournal      = "Advances in Mathematics",
 year          = "2011",
 volume        = "226",
 number        = "6",
 pages         = "4667--4701",
 eprint        = "1006.2117",
 coden         = "ADMTA4",
 issn          = "0001-8708",
 mrclass       = "15A18 (15A60 37B10 65K10 68R15)",
 mrnumber      = "2775881 (2012b:15016)",
 mrreviewer    = "Natalia Bebiano",
 zblnumber     = "1218.15005",
 zblreviewer   = "Rodica Covaci (Cluj-Napoca)",
 doi           = "10.1016/j.aim.2010.12.012",
 url           = "https://www.sciencedirect.com/science/article/pii/S0001870810004457",
 annote        = "The joint spectral radius of a finite set of real $d\times
                  d$ matrices is defined to be the maximum possible
                  exponential rate of growth of long products of matrices
                  drawn from that set. A set of matrices is said to have the
                  \emph{finiteness property} if there exists a periodic
                  product which achieves this maximal rate of growth.
                  \emph{J.\,C.~Lagarias} and
                  \emph{Y.~Wang}~\cite{LagWang:LAA95} conjectured in 1995
                  that every finite set of real $d \times d$ matrices
                  satisfies the finiteness property. However,
                  \emph{T.~Bousch} and \emph{J.~Mairesse}~\cite{BM:JAMS02}
                  proved in 2002 that counterexamples to the finiteness
                  conjecture exist, showing in particular that there exists a
                  family of pairs of $2\times 2$ matrices which contains a
                  counterexample. Similar results were subsequently given by
                  V.\,D.~Blondel, J.~Theys and A.\,A.~Vladimirov and by
                  V.\,S.~Kozyakin, but no explicit counterexample to the
                  finiteness conjecture has so far been given. The purpose of
                  this paper is to resolve this issue by giving the first
                  completely explicit description of a counterexample to the
                  Lagarias--Wang finiteness conjecture. Namely, for the set
                  \[ \mathsf{A}_{\alpha_*}:=
                  \left\{\left(\begin{array}{cc}1&1\\0&1\end{array}\right),
                  \alpha_*\left(\begin{array}{cc}1&0\\1&1\end{array}\right)\right\}
                  \] we give an explicit value of \[\alpha_* \simeq
                  0.749326546330367557943961948091344672091327370236064317358024\ldots
                  \] such that $\mathsf{A}_{\alpha_*}$ does not satisfy the
                  finiteness property.",
}

@ARTICLE{HMS:MPCPS13,
 author        = "Hare, Kevin G. and Morris, Ian D. and Sidorov, Nikita",
 authauthor    = "Hare, Kevin and Morris, Ian and Sidorov, Nikita",
 title         = "Extremal sequences of polynomial complexity",
 journal       = "Math. Proc. Cambridge Philos. Soc.",
 fjournal      = "Mathematical Proceedings of the Cambridge Philosophical
                  Society",
 year          = "2013",
 volume        = "155",
 number        = "2",
 pages         = "191--205",
 eprinttype    = "arXiv",
 eprint        = "1201.6236",
 issn          = "0305-0041",
 mrclass       = "15A60 (05A05)",
 mrnumber      = "3091514",
 mrreviewer    = "Olga Victorovna Markova",
 zblnumber     = "1326.15031",
 zblreviewer   = "Andreas Arvanitoyeorgos",
 doi           = "10.1017/S0305004113000157",
 url           = "https://dx.doi.org/10.1017/S0305004113000157",
 annote        = "The joint spectral radius of a bounded set of $d \times d$
                  real matrices is defined to be the maximum possible
                  exponential growth rate of products of matrices drawn from
                  that set. For a fixed set of matrices, a sequence of
                  matrices drawn from that set is called \emph{extremal} if
                  the associated sequence of partial products achieves this
                  maximal rate of growth. An influential conjecture of
                  \emph{J.~Lagarias} and \emph{Y.~Wang}~\cite{LagWang:LAA95}
                  asked whether every finite set of matrices admits an
                  extremal sequence which is periodic. This is equivalent to
                  the assertion that every finite set of matrices admits an
                  extremal sequence with bounded subword complexity.
                  Counterexamples were subsequently constructed which have
                  the property that every extremal sequence has at least
                  linear subword complexity. In this paper we extend this
                  result to show that for each integer $p\geq 1$, there
                  exists a pair of square matrices of dimension
                  $2^p(2^{p+1}-1)$ for which every extremal sequence has
                  subword complexity at least $2^{-p^2}n^p$.",
}

@BOOK{Hartfiel02,
 author        = "Hartfiel, Darald J.",
 authauthor    = "Hartfiel, Darald",
 title         = "Nonhomogeneous matrix products",
 publisher     = "World Scientific Publishing Co. Inc.",
 address       = "River Edge, NJ",
 year          = "2002",
 pages         = "x+224",
 isbn          = "9810246285",
 mrclass       = "15-02 (15A48 15A51 15A52)",
 mrnumber      = "1878339 (2002m:15001)",
 mrreviewer    = "E. Seneta",
 zblnumber     = "1011.15006",
 doi           = "10.1142/4707",
 annote        = "The book is a research exposition the aim of which is to put
                  together much of the basic work on nonhomogeneous matrix
                  products, i.e., $A_{k}\cdots A_{1}$ with (at least two)
                  different factors. The book consists of 13 chapters and an
                  appendix in which a few classical results used in the book
                  are given. Chapter~1 is an introduction containing some
                  preliminary definitions and remarks. In Chapter~2 the
                  projective metric in $\mathbb{R}^{n}$, Hausdorff metric and
                  other functionals that can be used to study convergence of
                  infinite products of matrices are explained. Chapter~3 is
                  devoted to algebraic properties and convergence in
                  semigroups of matrices. Chapter~4 deals with some
                  nonnegative matrices (i.e., having nonnegative entries) and
                  semigroups of such matrices. Ergodicity is studied in
                  Chapter~5. Here ergodicity concerns sequences of products
                  $A_{1}, A_{2}A_{1},\ldots\,$, which appear more like
                  rank~$1$ matrices as $k\to\infty$. In Chapter~6 some basic
                  convergence results for infinite products of matrices are
                  given, in Chapter~7 continuous convergence and some
                  sequence spaces are studied. A matrix $A$ is called
                  paracontracting (PC) if $\|Ax\|<\|x\|$ whenever $Ax\neq x$.
                  In Chapter~8 convergence of infinite products of PC
                  matrices is studied. In Chapters~9 and~10 a matrix set
                  $\Sigma$ is considered and the convergence in Hausdorff
                  metric of $\Sigma, \Sigma^{2},\ldots$ is studied. In
                  Chapter~12 slowly varying matrix products are considered.
                  In Chapters~11 and~13 some applications are shown. These
                  include, e.g., the construction of curves and fractals,
                  solving demographic problems, applications in production
                  systems and management structures. MATLAB codes for solving
                  some of those problems are given in the ends of Chapters~11
                  and~13.",
}

@ARTICLE{HC:JGO19,
 author        = "Heaton, Howard and Censor, Yair",
 title         = "Asynchronous sequential inertial iterations for common fixed
                  points problems with an application to linear systems",
 journal       = "J. Global Optim.",
 fjournal      = "Journal of Global Optimization. An International Journal
                  Dealing with Theoretical and Computational Aspects of
                  Seeking Global Optima and Their Applications in Science,
                  Management and Engineering",
 year          = "2019",
 volume        = "74",
 number        = "1",
 pages         = "95--119",
 eprinttype    = "arXiv",
 eprint        = "1808.04723",
 issn          = "0925-5001",
 mrclass       = "47J25 (47H10 90C25 90C48)",
 mrnumber      = "3943617",
 zblnumber     = "07069296",
 doi           = "10.1007/s10898-019-00747-4",
 url           = "https://link.springer.com/article/10.1007/s10898-019-00747-4",
 annote        = "The common fixed points problem requires finding a point in
                  the intersection of fixed points sets of a finite
                  collection of operators. Quickly solving problems of this
                  sort is of great practical importance for engineering and
                  scientific tasks (e.g., for computed tomography). Iterative
                  methods for solving these problems often employ a
                  Krasnosel'ski{\u\i}-Mann type iteration. We present an
                  Asynchronous Sequential Inertial (ASI) algorithmic
                  framework in a Hilbert space to solve common fixed points
                  problems with a collection of nonexpansive operators. Our
                  scheme allows use of out-of-date iterates when generating
                  updates, thereby enabling processing nodes to work
                  simultaneously and without synchronization. This method
                  also includes inertial type extrapolation terms to increase
                  the speed of convergence. In particular, we extend the
                  application of the recent ``ARock algorithm'' [Peng, Z. et
                  al, SIAM J. on Scientific Computing {\bf 38}, A2851-2879,
                  (2016)] in the context of convex feasibility problems.
                  Convergence of the ASI algorithm is proven with no
                  assumption on the distribution of delays, except that they
                  be uniformly bounded. Discussion is provided along with a
                  computational example showing the performance of the ASI
                  algorithm applied in conjunction with a diagonally relaxed
                  orthogonal projections (DROP) algorithm for estimating
                  solutions to large linear systems.",
}

@INCOLLECTION{HStr:LASP92,
 author        = "Heil, Christopher and Strang, Gilbert",
 title         = "Continuity of the joint spectral radius: application to
                  wavelets",
 booktitle     = "Linear algebra for signal processing ({M}inneapolis, {MN},
                  1992)",
 series        = "IMA Vol. Math. Appl.",
 publisher     = "Springer",
 address       = "New York",
 year          = "1995",
 volume        = "69",
 pages         = "51--61",
 mrclass       = "15A60 (42C15 94A12)",
 mrnumber      = "1351732 (96h:15028)",
 mrreviewer    = "Zden{\v{e}}k Dost{\'a}l",
 zblnumber     = "0823.15009",
 doi           = "10.1007/978-1-4612-4228-4_4",
 annote        = "The joint spectral radius is the extension to two or more
                  matrices of the (ordinary) spectral radius
                  $\rho(A)=\max|\lambda_{i}(A)| = \lim \| A_{m}\|^{1/m}$. The
                  extension allows matrix products $\prod_{m}$ taken in all
                  orders, so that norms and eigenvalues are difficult to
                  estimate. We show that the limiting process does yield a
                  continuous function of the original matrices -- this is
                  their joint spectral radius. Then we describe the
                  construction of wavelets from a dilation equation with
                  coefficients $c_{k}$. We connect the continuity of those
                  wavelets to the value of the joint spectral radius of two
                  matrices whose entries are formed from the $c_{k}$.",
}

@ARTICLE{Holtz:EJLA00,
 author        = "Holtz, Olga",
 title         = "On convergence of infinite matrix products",
 journal       = "Electron. J. Linear Algebra",
 fjournal      = "Electronic Journal of Linear Algebra",
 year          = "2000",
 volume        = "7",
 pages         = "178--181",
 eprinttype    = "arXiv",
 eprint        = "math/0512590",
 issn          = "1081-3810",
 mrclass       = "15A60",
 mrnumber      = "1781470 (2001c:15036)",
 zblnumber     = "0964.15030",
 zblreviewer   = "Mihail Voicu (Ia{\c{s}}i)",
 url           = "https://journals.uwyo.edu/index.php/ela/article/view/109",
 annote        = "The author shows that the convergence of the product
                  \[P=\prod_{n\in N}\left[\begin{array}{cc}I&B_n\\0&C_n
                  \end{array}\right],\] where $C_n$ are (uniform) contracting
                  submatrices, is equivalent with the convergence of the
                  subsequence $X_{n}=B_{n}(I-C_{n})^{-1}$ and \[P=\prod_{n\in
                  N}\left[\begin{array}{cc}I&\lim\limits_{n\to\infty}X_{n}\\0&0
                  \end{array}\right].\]",
}

@UNPUBLISHED{Howard97,
 author        = "Howard, Ralph",
 title         = "The {J}ohn Ellipsoid Theorem",
 year          = "1997",
 url           = "https://api.semanticscholar.org/CorpusID:117097043",
 annote        = "This is a lecture given in the functional analysis seminar
                  at the University of South Carolina. Contents: 1.
                  Introduction. 2. Proof of the theorem. 3. The case of
                  centrally symmetric convex bodies. 4. Proof of the
                  uniquness of the John ellipsoid. 4.1. Examples of where the
                  inequalities are sharp.",
}

@ARTICLE{ShihHsu:JNCA11,
 author        = "Hsu, Sheng-Yi and Shih, Mau-Hsiang",
 title         = "A proof of {F}ritz {J}ohn ellipsoid theorem",
 journal       = "J. Nonlinear Convex Anal.",
 fjournal      = "Journal of Nonlinear and Convex Analysis. An International
                  Journal",
 year          = "2011",
 volume        = "12",
 number        = "1",
 pages         = "1--4",
 issn          = "1345-4773",
 mrclass       = "15A60 (52A20)",
 mrnumber      = "2816404 (2012c:15048)",
 zblnumber     = "1215.15026",
 annote        = "We present a simple proof of {\it Fritz John}'s ellipsoid
                  theorem [cf. Fritz John. Collected papers. Volumes 1, 2.
                  Contemporary Mathematicians, 543--560 (1985)] using a
                  projection theorem proved by the Hahn--Banach theorem.",
}

@ARTICLE{HSZ:IEEETAC09,
 author        = "Hu, Jianghai and Shen, Jinglai and Zhang, Wei",
 title         = "Generating functions of switched linear systems: {A}nalysis,
                  computation, and stability applications",
 journal       = "IEEE Trans. Automat. Control",
 fjournal      = "Institute of Electrical and Electronics Engineers.
                  Transactions on Automatic Control",
 year          = "2011",
 volume        = "56",
 number        = "5",
 pages         = "1059--1074",
 coden         = "IETAA9",
 issn          = "0018-9286",
 mrclass       = "93C30 (93C55 93D20)",
 mrnumber      = "2815911 (2012e:93079)",
 mrreviewer    = "Felipe Miguel Pait",
 zblnumber     = "1368.93281",
 doi           = "10.1109/TAC.2010.2067590",
 url           = "https://ieeexplore.ieee.org/document/5549855",
 annote        = "In this paper, a unified framework is proposed to study the
                  exponential stability of discrete-time switched linear
                  systems and, more generally, the exponential growth rates
                  of their trajectories under three types of switching rules:
                  arbitrary switching, optimal switching, and random
                  switching. It is shown that the maximum exponential growth
                  rates of system trajectories over all initial states under
                  these three switching rules are completely characterized by
                  the radii of convergence of three suitably defined families
                  of functions called the strong, the weak, and the mean
                  generating functions, respectively. In particular,
                  necessary and sufficient conditions for the exponential
                  stability of the switched linear systems are derived based
                  on these radii of convergence. Various properties of the
                  generating functions are established, and their relations
                  are discussed. Algorithms for computing the generating
                  functions and their radii of convergence are also developed
                  and illustrated through examples.",
}

@ARTICLE{HLHX:JMAA11,
 author        = "Huang, Yu and Luo, Jun and Huang, Tingwen and Xiao,
                  MingQing",
 authauthor    = "Huang, Yu and Luo, Jun and Huang, Tingwen and Xiao,
                  MingQing",
 title         = "The set of stable switching sequences for discrete-time
                  linear switched systems",
 journal       = "J. Math. Anal. Appl.",
 fjournal      = "Journal of Mathematical Analysis and Applications",
 year          = "2011",
 volume        = "377",
 number        = "2",
 pages         = "732--743",
 coden         = "JMANAK",
 issn          = "0022-247X",
 mrclass       = "93D20 (28D05 39A30 60G17 93C30 93C55)",
 mrnumber      = "2769170 (2011m:93148)",
 mrreviewer    = "V{\~u} Ngoc Ph{\'a}t",
 zblnumber     = "1214.93089",
 doi           = "10.1016/j.jmaa.2010.11.053",
 url           = "https://www.sciencedirect.com/science/article/pii/S0022247X10009881",
 annote        = "We study the characterization of the asymptotical stability
                  for discrete-time switched linear systems. We first
                  translate the system dynamics into a symbolic setting under
                  the framework of symbolic topology. Then by using the
                  ergodic measure theory, a lower bound estimate of Hausdorff
                  dimension of the set of asymptotically stable sequences is
                  obtained. We show that the Hausdorff dimension of the set
                  of asymptotically stable switching sequences is positive if
                  and only if the corresponding switched linear system has at
                  least one asymptotically stable switching sequence. The
                  obtained result reveals an underlying fundamental
                  principle: a switched linear system either possesses
                  uncountable numbers of asymptotically stable switching
                  sequences or has none of them, provided that the switching
                  is arbitrary. We also develop frequency and density indexes
                  to identify those asymptotically stable switching sequences
                  of the system.",
}

@ARTICLE{Jachymski:TAMS09,
 author        = "Jachymski, Jacek",
 title         = "K{\"o}nig chains for submultiplicative functions and
                  infinite products of operators",
 journal       = "Trans. Amer. Math. Soc.",
 fjournal      = "Transactions of the American Mathematical Society",
 year          = "2009",
 volume        = "361",
 number        = "11",
 pages         = "5967--5981",
 issn          = "0002-9947",
 mrclass       = "47A35 (15A60 26B35 46H05 47B38 54D20 54D30)",
 mrnumber      = "2529921",
 zblnumber     = "1195.47009",
 zblreviewer   = "Peter Lappan (East Lansing)",
 doi           = "10.1090/S0002-9947-09-04909-5",
 url           = "https://www.ams.org/journals/tran/2009-361-11/S0002-9947-09-04909-5/",
 annote        = "Let $\Sigma$ be a set and let $\Sigma^{n}$ (resp.,
                  $\Sigma^{\infty}$) denote the Cartesian product of $n$
                  (resp., $\infty$) copies of $\Sigma$. Let
                  $P_{(1,2,\ldots,n)}$ denote the projection of
                  $\Sigma^{\infty}$ onto $\Sigma^{n}$, that is,
                  \[P_{(1,2,\ldots,n)}(\sigma_{1},\sigma_{2},\ldots)=
                  (\sigma_{1},\sigma_{2},\ldots,\sigma_{n} ),\qquad
                  \sigma=(\sigma_{1},\sigma_{2},\ldots)\in\Sigma^{\infty}.\]
                  The shift operator $s:\Sigma^{\infty}\to\Sigma^{\infty}$ is
                  defined as
                  $s(\sigma_{1},\sigma_{2},\sigma_{3},\ldots)=(\sigma_{2},\sigma_{3},\ldots)$.
                  A subset $K$ of $\Sigma^{\infty}$ is said to be shift
                  invariant if $s(K)\subset K$. For a shift invariant set
                  $K\subset\Sigma^{\infty}$, let
                  $B_{n}(K)=P_{(1,2,\ldots,n)}(K)$ for each positive integer
                  $n$, and let $B(K)=\bigcup_{n}B_{n}(K)$. A function
                  non-negative $\Phi$ on $B(K)$ is said to be
                  submultiplicative if
                  \[\Phi(\sigma_{1},\sigma_{2},\ldots,\sigma_{n})\leq
                  \Phi(\sigma_{1},\sigma_{2},\ldots,\sigma_{j})
                  \Phi(\sigma_{j+1},\ldots,\sigma_{n}),\qquad 1\leq j\leq
                  n-1,\] for each $n\geq 2$ and
                  $(\sigma_{1},\sigma_{2},\ldots,\sigma_{n})\in B_{n}(K)$. A
                  sequence $(\sigma_{1},\sigma_{2},\ldots)\in\Sigma^{\infty}$
                  is said to be a K{\"o}nig chain for
                  $\{X_{n}:n=1,2,\ldots\}$ if
                  $(\sigma_{1},\ldots,\sigma_{n})\in X_{n}$ for
                  $n=1,2,\ldots$, where $X_{n}\subset\Sigma^{n}$ for each
                  positive integer $n$. The author gives some conditions
                  under which there exists a K{\"o}nig chain for a sequence
                  of sets $\{X_{n}\}$, and also describes some situations for
                  which a sequence $\{X_{n}\}$ does not have K{\"o}nig
                  chains. For a submultiplicative function $\Phi$, let
                  $\rho_{n}(\Phi)=\sup\{\Phi(w):w\in B_{n}(K)\}$, where
                  $K\subset\Sigma^{\infty}$ is shift invariant, then
                  \[\rho(\Phi)=\lim_{n\to \infty}\rho_{n}^{1/n}(\Phi)\] is
                  called the radius of $\Phi$. An element
                  $(\sigma_{1},\sigma_{2},\ldots)\in K\subset\Sigma^{\infty}$
                  is called a K{\"o}nig chain for $\Phi$ if
                  $\Phi(\sigma_{1},\sigma_{2},\ldots,\sigma_{n})\geq
                  (\rho(\Phi))^{n}$ for each positive integer $n$. The author
                  proves that, if $\Sigma$ is a compact topological space,
                  $K$ is a nonempty closed and shift invariant subset of
                  $\Sigma^{\infty}$, and if $\Phi$ is a positive real valued
                  submultiplicative function on $B(K)$ such that
                  $\Phi|_{B_{n}(K)}$ is upper semicontinuous, then there
                  exists a K{\"o}nig chain for $\Phi$. This result leads to
                  an improvement of a result of
                  \emph{L.~Mate}~\cite{Mate:FM99}. The author also
                  investigates \emph{Hutchinson systems} of self-maps of a
                  metric space, and he obtains an extension of a result due
                  to \emph{R.\,P.~Kelisky} and \emph{T.\,J.~Rivlin} [Pac. J.
                  Math. 21, 511--520 (1967)] concerning iterates of the
                  Bernstein operators on the Banach space $C([0,1])$.",
}

@ARTICLE{Javaheri:JMAA13,
 author        = "Javaheri, Mohammad",
 title         = "Maximally transitive semigroups of {$n\times n$} matrices",
 journal       = "J. Math. Anal. Appl.",
 fjournal      = "Journal of Mathematical Analysis and Applications",
 year          = "2013",
 volume        = "401",
 number        = "2",
 pages         = "743--753",
 coden         = "JMANAK",
 issn          = "0022-247X",
 mrclass       = "15A30 (20M20)",
 mrnumber      = "3018024",
 zblnumber     = "06156283",
 doi           = "10.1016/j.jmaa.2012.12.047",
 url           = "https://www.sciencedirect.com/science/article/pii/S0022247X12010384",
 annote        = "We prove that, for every $n\ge 1$, there exists a pair of
                  $n\times n$ matrices that generates a topologically
                  $n$-transitive semigroup action on $\mathbb{K}^n$, where
                  $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$. Equivalently, we
                  construct dense $2$-generator subsemigroups of
                  $GL(n,\mathbb{K})$ for all $n\ge 1$.",
}

@ARTICLE{JenPoll:ETDS17,
 author        = "Jenkinson, Oliver and Pollicott, Mark",
 title         = "Joint spectral radius, {S}turmian measures, and the
                  finiteness conjecture",
 journal       = "Ergodic Theory Dynam. Systems",
 fjournal      = "Ergodic Theory and Dynamical Systems",
 publisher     = "Cambridge University Press",
 year          = "2017",
 pages         = "1--39",
 eprinttype    = "arXiv",
 eprint        = "1501.03419",
 issn          = "0143-3857",
 mrclass       = "15A60",
 mrnumber      = "3868023",
 zblnumber     = "1405.15028",
 zblreviewer   = "Jorma K. Merikoski",
 doi           = "10.1017/etds.2017.18",
 url           = "https://dx.doi.org/10.1017/etds.2017.18",
 annote        = "The joint spectral radius of a pair of $2 \times 2$ real
                  matrices $(A_0,A_1)\in M_2(\mathbb{R})^2$ is defined to be
                  $r(A_0,A_1)= \limsup_{n\to\infty} \max \{\|A_{i_1}\cdots
                  A_{i_n}\|^{1/n}: i_j\in\{0,1\}\}$, the optimal growth rate
                  of the norm of products of these matrices. The
                  Lagarias--Wang finiteness conjecture~\cite{LagWang:LAA95},
                  asserting that $r(A_0,A_1)$ is always the $n$th root of the
                  spectral radius of some length-$n$ product $A_{i_1}\cdots
                  A_{i_n}$, has been refuted by \emph{T.~Bousch} \&
                  \emph{J.~Mairesse}~\cite{BM:JAMS02}, with subsequent
                  counterexamples presented by \emph{V.~Blondel},
                  \emph{J.~Theys} and
                  \emph{A.~Vladimirov}~\cite{BTV:SIAMJMA03},
                  \emph{V.~Kozyakin}~\cite{Koz:CDC05:e}, \emph{K.~Hare},
                  \emph{I.~Morris}, \emph{N.~Sidorov} and
                  \emph{J.~Theys}~\cite{HMST:AdvMath11}.\par In this article
                  we present a large class of finiteness counterexamples,
                  proving that there exists an open subset of
                  $M_2(\mathbb{R})^2$ with the property that each member
                  $(A_0,A_1)$ of the subset generates uncountably many
                  counterexamples of the form $(A_0,tA_1)$. Our methods
                  employ ergodic theory, in particular the analysis of
                  Sturmian invariant measures; this approach allows a short
                  proof that the relation between the parameter $t$ and the
                  Sturmian parameter $\mathcal{P}(t)$ is a Devil's
                  staircase.",
}

@ARTICLE{Jia:ACM95,
 author        = "Jia, Rong-Qing",
 title         = "Subdivision schemes in {$L_p$} spaces",
 journal       = "Adv. Comput. Math.",
 fjournal      = "Advances in Computational Mathematics",
 year          = "1995",
 volume        = "3",
 number        = "4",
 pages         = "309--341",
 issn          = "1019-7168",
 mrclass       = "65D15 (41A30)",
 mrnumber      = "1339166 (96d:65028)",
 mrreviewer    = "Dietrich Braess",
 zblnumber     = "0833.65148",
 doi           = "10.1007/BF03028366",
 url           = "https://link.springer.com/article/10.1007/BF03028366",
 annote        = "Subdivision schemes play an important role in computer
                  graphics and wavelet analysis. In this paper we are mainly
                  concerned with convergence of subdivision schemes in $L_p$
                  spaces $(1\le p \le\infty)$. We characterize the
                  $L_p$-convergence of a subdivision scheme in terms of the
                  $p$-norm joint spectral radius of two matrices associated
                  with the corresponding mask. We also discuss various
                  properties of the limit function of a subdivision scheme,
                  such as stability, linear independence, and smoothness.",
}

@ARTICLE{JLZ:JFAA01,
 author        = "Jia, Rong-Qing and Lau, Ka-Sing and Zhou, Ding-Xuan",
 title         = "{$L_{p}$} solutions of refinement equations",
 journal       = "J. Fourier Anal. Appl.",
 fjournal      = "The Journal of Fourier Analysis and Applications",
 year          = "2001",
 volume        = "7",
 number        = "2",
 pages         = "143--167",
 issn          = "1069-5869",
 mrclass       = "42C40",
 mrnumber      = "1817673 (2002i:42049)",
 zblnumber     = "1030.42031",
 doi           = "10.1007/BF02510421",
 url           = "https://link.springer.com/article/10.1007/BF02510421",
 annote        = "In the recent characterizations of the $L_p$-solution of the
                  refinement equation in terms of the `$p$-norm joint
                  spectral radius', there are problems in choosing the
                  initial function for iteration, and, in addition, in
                  requiring stability of the refinable function. In this
                  paper, we overcome these difficulties and give a more
                  complete characterization of this nature. The criterion is
                  constructive and can be implemented. It can be used to
                  describe the regularity of the solution without assuming
                  stability. This has significant advantages over the
                  previous work. The corresponding results for vector
                  refinement equations are also discussed.",
}

@INCOLLECTION{JohnF:48,
 author        = "John, Fritz",
 title         = "Extremum problems with inequalities as subsidiary
                  conditions",
 booktitle     = "Studies and {E}ssays {P}resented to {R}. {C}ourant on his
                  60th {B}irthday, {J}anuary 8, 1948",
 publisher     = "Interscience Publishers, Inc., New York, N. Y.",
 year          = "1948",
 pages         = "187--204",
 mrclass       = "49.0X",
 mrnumber      = "0030135 (10,719b)",
 mrreviewer    = "J. E. Wilkins, Jr.",
 zblnumber     = "0034.10503",
 annote        = "This paper deals with an extension of Lagrange's multiplier
                  rule to the case, where the subsidiary conditions are
                  inequalities instead of equations. Only extrema of
                  differentiable functions of a finite number of variables
                  will be considered. There may however be an infinite number
                  of inequalities prescribed. Lagrange's rule for the
                  situation considered here differs from the ordinary one, in
                  that the multipliers may always be assumed to be positive.
                  This makes it possible to obtain sufficient conditions for
                  the occurence or a minimum in terms of the first
                  derivatives only. See also~\cite{JohnF:2014}.",
}

@INCOLLECTION{JohnF:2014,
 author        = "John, Fritz",
 editor        = "Giorgi, Giorgio and Kjeldsen, Tinne Hoff",
 title         = "Extremum problems with inequalities as subsidiary
                  conditions",
 booktitle     = "Traces and Emergence of Nonlinear Programming",
 publisher     = "Birkh{\"a}user/Springer Basel AG",
 address       = "Basel",
 year          = "2014",
 pages         = "197--215",
 isbn          = "978-3-0348-0439-4",
 mrclass       = "90-03 (01A75 90C30 90C46)",
 mrnumber      = "3204131",
 doi           = "10.1007/978-3-0348-0439-4_9",
 url           = "https://link.springer.com/chapter/10.1007/978-3-0348-0439-4_9",
 note          = "reprint of~\cite{JohnF:48}",
 annote        = "This paper deals with an extension of Lagrange's multiplier
                  rule to the case, where the subsidiary conditions are
                  inequalities instead of equations. Only extrema of
                  differentiable functions of a finite number of variables
                  will be considered. There may however be an infinite number
                  of inequalities prescribed. Lagrange's rule for the
                  situation considered here differs from the ordinary one, in
                  that the multipliers may always be assumed to be positive.
                  This makes it possible to obtain sufficient conditions for
                  the occurence or a minimum in terms of the first
                  derivatives only.",
}

@ARTICLE{JP:SIAMJMA09,
 author        = "Jungers, R. M. and Protasov, V. {\relax{}Yu}.",
 authauthor    = "Jungers, R. and Protasov, V.",
 title         = "Counterexamples to the complex polytope extremality
                  conjecture",
 journal       = "SIAM J. Matrix Anal. Appl.",
 fjournal      = "SIAM Journal on Matrix Analysis and Applications",
 year          = "2009",
 volume        = "31",
 number        = "2",
 pages         = "404--409",
 issn          = "0895-4798",
 mrclass       = "93D09 (15A18)",
 mrnumber      = "2530256 (2011a:93129)",
 mrreviewer    = "Nicola Guglielmi",
 zblnumber     = "1202.15027",
 zblreviewer   = "Alexey Alimov (Moskva)",
 doi           = "10.1137/080730652",
 url           = "https://epubs.siam.org/doi/10.1137/080730652",
 annote        = "The authors disprove a~recent conjecture due to
                  \emph{N.~Guglielmi}, \emph{F.~Wirth}, and
                  \emph{M.~Zennaro}~\cite{GWZ:SIAMJMA05} stating that any
                  nondefective set of matrices having the finiteness property
                  has an extremal complex polytope norm. The main result to
                  disprove this conjecture is the following:\par
                  \textbf{Theorem 4.} \emph{There exists an~irreducible pair
                  of $3\times 3$ orthogonal matrices $A_0, A_1$ for which
                  there is no complex polytope $P\subset \mathbb{C}^3$ such
                  that $A_iP\subset P$, $i=0,1$.}\par \emph{Also, there
                  exists a~nondefective pair of $3\times 3$ matrices
                  $A_0$,~$A_1$ with nonnegative entries for which
                  $\rho(\{A_0,A_1\}) =\rho(A_0)=\rho(A_1)=1$, but there is no
                  nondegenerate complex polytope $P\subset \mathbb{C}^3$ such
                  that $A_iP\subset P$, $i=0,1$.}\par Here $\rho(\cdot)$ is
                  the joint spectral radius.\par The authors give two
                  counterexamples to show that the conjecture is false even
                  if the set of matrices is supposed to admit the positive
                  orthant as an invariant cone, or even if the set of
                  matrices is assumed to be irreducible.",
}

@INPROCEEDINGS{JP:CDC10,
 author        = "Jungers, R. M. and Protasov, V. {\relax{}Yu}.",
 authauthor    = "Jungers, R. and Protasov, V.",
 title         = "Weak stability of switching dynamical systems and fast
                  computation of the {$p$}-radius of matrices",
 booktitle     = "Proceedings of 49th IEEE Conference on Decision and Control
                  (CDC)",
 address       = "Atlanta, GA",
 year          = "2010",
 pages         = "7328--7333",
 doi           = "10.1109/CDC.2010.5717653",
 url           = "https://ieeexplore.ieee.org/document/5717653",
 annote        = "The stability of a switching linear dynamical system is
                  ruled by the so-called joint spectral radius of the set of
                  matrices characterizing the dynamical system. In some
                  situations, the system is not stable in the classical
                  sense, but might still be stable in a weaker meaning. We
                  introduce the new notion of weak stability or
                  $L_p$-stability of a switched dynamical system based on the
                  so-called $p$-radius of the set of matrices. The $p$-radius
                  characterizes the average rate of growth of norms of
                  matrices in a multiplicative semigroup. This quantity has
                  found several applications in the recent years. We analyze
                  the computability of this quantity, and we describe a
                  series of approximations that converge to the $p$-radius
                  with a priori computable accuracy. For nonnegative
                  matrices, this gives efficient approximation schemes for
                  the $p$-radius computation. We finally show the efficiency
                  of our methods on several practical examples.",
}

@BOOK{Jungers:09,
 author        = "Jungers, Rapha{\"e}l",
 title         = "The joint spectral radius",
 series        = "Lecture Notes in Control and Information Sciences",
 publisher     = "Springer-Verlag",
 address       = "Berlin",
 year          = "2009",
 volume        = "385",
 pages         = "xiv+144",
 isbn          = "978-3-540-95979-3",
 mrclass       = "15-02 (05A05 05C50 15A18 15A60 42C40)",
 mrnumber      = "2507938 (2011c:15001)",
 mrreviewer    = "Nicola Guglielmi",
 doi           = "10.1007/978-3-540-95980-9",
 url           = "https://link.springer.com/book/10.1007/978-3-540-95980-9",
 note          = "{T}heory and applications",
 annote        = "This monograph is based on the Ph.D. Thesis of the author.
                  Its goal is twofold:\par First, it presents most research
                  work that has been done during his Ph.D., or at least the
                  part of the work that is related with the joint spectral
                  radius. This work was concerned with theoretical
                  developments (part I) as well as the study of some
                  applications (part II). As a second goal, it was the
                  author's feeling that a survey on the state of the art on
                  the joint spectral radius was really missing in the
                  literature, so that the first two chapters of part I
                  present such a survey. The other chapters mainly report
                  personal research, except Chapter 5 which presents an
                  important application of the joint spectral radius: the
                  continuity of wavelet functions.\par The first part of this
                  monograph is dedicated to theoretical results. The first
                  two chapters present the above mentioned survey on the
                  joint spectral radius. Its minimum-growth counterpart, the
                  \emph{joint spectral subradius}, is also considered. The
                  next two chapters point out two specific theoretical
                  topics, that are important in practical applications: the
                  particular case of nonnegative matrices, and the Finiteness
                  Property.\par The second part considers applications
                  involving the joint spectral radius. We first present the
                  continuity of wavelets. We then study the problem of the
                  capacity of codes submitted to forbidden difference
                  constraints. Then we go to the notion of overlap-free
                  words, a problem that arises in combinatorics on words. We
                  then end with the problem of trackability of sensor
                  networks, and show how the theoretical results developed in
                  the first part allow to solve this problem efficiently.",
}

@ARTICLE{Jungers:LAA12,
 author        = "Jungers, Rapha{\"e}l M.",
 authauthor    = "Jungers, Rapha{\"e}l",
 title         = "On asymptotic properties of matrix semigroups with an
                  invariant cone",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2012",
 volume        = "437",
 number        = "5",
 pages         = "1205--1214",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15B48 (15A60 65F15 93C30)",
 mrnumber      = "2942343",
 mrreviewer    = "Olga Y. Kushel",
 zblnumber     = "1258.15015",
 zblreviewer   = "Valeriu Prepeli{\c{t}}{\u{a}} (Bucure{\c{s}}ti)",
 doi           = "10.1016/j.laa.2012.04.006",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379512002704",
 annote        = "Two joint spectral characteristics are studied for finitely
                  generated matrix semigroups, when the matrices share one or
                  two invariant cones. For a bounded set
                  $\Sigma\in\mathbb{R}^{n\times n}$, these characteristics
                  are the joint spectral radius $\rho(\Sigma)$ and the joint
                  spectral subradius $\check{\rho}(\Sigma)$ respectively,
                  \[\rho(\Sigma)=\lim_{t\rightarrow\infty}\sup\{\|A_1\cdots
                  A_t\|^{1/t}: A_i\in\Sigma\},\
                  \check{\rho}(\Sigma)=\lim_{t\rightarrow\infty}\inf\{\|
                  A_1\cdots A_t\|^{1/t}: A_i\in\Sigma\}.\] It is well-known
                  that $\rho(\Sigma)$ is continuous w.r.t. the Hausdorff
                  distance, but $\check{\rho}(\Sigma)$ is not continuous. In
                  this paper the continuity of the joint spectral subradius
                  is proved in the neighborhood of sets of matrices that
                  leave an embedded pair of cones invariant.\par A convex
                  closed cone $K'$ is embedded in a cone
                  $K\subset\mathbb{R}^{n\times n}$ if
                  $(K'\setminus\{0\})\subset\text {int} K$ and in this case
                  one says that $\{K,K'\}$ is an embedded pair. The main
                  theorem is the following:\par \textbf{Theorem.} \emph{If
                  $\Sigma$ is a compact set in $\mathbb{R}^{n\times n}$ which
                  leaves an embedded pair of cones invariant and $(\Sigma_k)$
                  is a sequence of sets in $\mathbb{R}^{n\times n}$ that
                  converges to $\Sigma$ in the Hausdorff metric, then
                  $\check{\rho}(\Sigma_k)\rightarrow\check{\rho}(\Sigma)$ as
                  $k\rightarrow\infty$.}\par The author denotes by $\Sigma^t$
                  the set of products of length $t$ of matrices from the
                  bounded set $\Sigma$ of matrices which leave a cone $K$
                  invariant. He proves that if there exists $A\in \Sigma$
                  which is $K$-primitive (i.e. $\exists m\in\mathbb{N}$ such
                  that $A^m(K\setminus\{0\})\subset\text{int} K$), then both
                  the maximal trace
                  $\displaystyle\max_{A\in\Sigma^t}\{\text{tr}^{1/t}(A)\}$
                  and the averaged maximal spectral radius
                  $\displaystyle\max_{A\in\Sigma^t}\{\rho^{1/t}(A)\}$
                  converge to the joint spectral radius as
                  $t\rightarrow\infty$.",
}

@INCOLLECTION{Jungers:LNCS13,
 author        = "Jungers, Rapha{\"e}l M.",
 authauthor    = "Jungers, Rapha{\"e}l",
 title         = "Joint spectral characteristics: a tale of three
                  disciplines",
 booktitle     = "Developments in language theory. Proceedings of the 17th
                  international conference, DLT 2013, Marne-la-Vall{\'e}e,
                  France, June 18--21, 2013",
 series        = "Lecture Notes in Computer Science",
 publisher     = "Springer Berlin Heidelberg",
 year          = "2013",
 volume        = "7907",
 pages         = "27--28",
 isbn          = "978-3-642-38770-8/pbk",
 mrclass       = "68Q45",
 doi           = "10.1007/978-3-642-38771-5_3",
 url           = "https://link.springer.com/chapter/10.1007/978-3-642-38771-5_3",
 annote        = "Joint spectral characteristics describe the stationary
                  behavior of a discrete time linear switching system. Well,
                  that's what an electrical engineer would say. A
                  mathematician would say that they characterize the
                  asymptotic behavior of a semigroup of matrices, and a
                  computer scientist would perhaps see them as describing
                  languages generated by automata.",
}

@ARTICLE{JB:LAA08,
 author        = "Jungers, Rapha{\"e}l M. and Blondel, Vincent D.",
 authauthor    = "Jungers, Rapha{\"e}l and Blondel, Vincent",
 title         = "On the finiteness property for rational matrices",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2008",
 volume        = "428",
 number        = "10",
 pages         = "2283--2295",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15A18 (05B20 15A36 15A60)",
 mrnumber      = "2405245 (2009e:15029)",
 mrreviewer    = "Alan L. Andrew",
 zblnumber     = "1148.15004",
 zblreviewer   = "Jinhai Chen (Hongkong)",
 doi           = "10.1016/j.laa.2007.07.007",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379507003138",
 annote        = "We analyze the periodicity of optimal long products of
                  matrices. A set of matrices is said to have the finiteness
                  property if the maximal rate of growth of long products of
                  matrices taken from the set can be obtained by a periodic
                  product. It was conjectured a decade ago that all finite
                  sets of real matrices have the finiteness property. This
                  ``finiteness conjecture'' is now known to be false but no
                  explicit counterexample is available and in particular it
                  is unclear if a counterexample is possible whose matrices
                  have rational or binary entries. In this paper, we prove
                  that all finite sets of nonnegative rational matrices have
                  the finiteness property if and only if \emph{pairs} of
                  \emph{binary matrices} do and we state a similar result
                  when negative entries are allowed. We also show that all
                  pairs of $2\times 2$ binary matrices have the finiteness
                  property. These results have direct implications for the
                  stability problem for sets of matrices. Stability is
                  algorithmically decidable for sets of matrices that have
                  the finiteness property and so it follows from our results
                  that if all pairs of binary matrices have the finiteness
                  property then stability is decidable for nonnegative
                  rational matrices. This would be in sharp contrast with the
                  fact that the related problem of boundedness is known to be
                  undecidable for sets of nonnegative rational matrices.",
}

@ARTICLE{JGC:SIAMJMAA14,
 author        = "Jungers, Rapha{\"e}l M. and Cicone, Antonio and Guglielmi,
                  Nicola",
 authauthor    = "Rapha{\"e}l Jungers and Nicola Guglielmi and Antonio
                  Cicone",
 title         = "Lifted polytope methods for computing the joint spectral
                  radius",
 journal       = "SIAM J. Matrix Anal. Appl.",
 fjournal      = "SIAM Journal on Matrix Analysis and Applications",
 year          = "2014",
 volume        = "35",
 number        = "2",
 pages         = "391--410",
 eprinttype    = "arXiv",
 eprint        = "1207.5123",
 issn          = "0895-4798",
 mrclass       = "15A60 (37C75 65F15 90C22)",
 mrnumber      = "3188391",
 mrreviewer    = "Wen Li",
 zblnumber     = "1296.93067",
 doi           = "10.1137/130907811",
 url           = "https://epubs.siam.org/doi/10.1137/130907811",
 annote        = "We describe new methods for deciding the stability of
                  switching systems. The methods build on two ideas
                  previously appeared in the literature: the polytope norm
                  iterative construction, and the lifting procedure.
                  Moreover, the combination of these two ideas allows us to
                  introduce a pruning algorithm which can importantly reduce
                  the computational burden. We prove several appealing
                  theoretical properties of our methods like a finiteness
                  computational result which extends a known result for
                  unlifted sets of matrices, and provide numerical examples
                  of their good behaviour.",
}

@ARTICLE{JunMas:SIAMJCO17,
 author        = "Jungers, Rapha{\"e}l M. and Mason, Paolo",
 authauthor    = "Jungers, Rapha{\"e}l and Mason, Paolo",
 title         = "On feedback stabilization of linear switched systems via
                  switching signal control",
 journal       = "SIAM J. Control Optim.",
 fjournal      = "SIAM Journal on Control and Optimization",
 year          = "2017",
 volume        = "55",
 number        = "2",
 pages         = "1179--1198",
 eprinttype    = "arXiv",
 eprint        = "1601.08141",
 issn          = "0363-0129",
 mrclass       = "93D15 (93C30 93D30)",
 mrnumber      = "3633780",
 mrreviewer    = "Nikolaos Athanasopoulos",
 zblnumber     = "1361.93051",
 doi           = "10.1137/15M1027802",
 url           = "https://epubs.siam.org/doi/10.1137/15M1027802",
 annote        = "Motivated by recent applications in control theory, we study
                  the feedback stabilizability of switched systems, where one
                  is allowed to chose the switching signal as a function of
                  $x(t)$ in order to stabilize the system. We propose new
                  algorithms and analyze several mathematical features of the
                  problem which were unnoticed up to now, to our knowledge.
                  We prove complexity results, (in)equivalence between
                  various notions of stabilizability, existence of Lyapunov
                  functions, and we provide a case study for a paradigmatic
                  example introduced by \emph{D.\,P.~Stanford} and
                  \emph{J.\,M.~Urbano}~\cite{SU:SIAMJMAA94}.",
}

@ARTICLE{JPB:LAA08,
 author        = "Jungers, Rapha{\"e}l M. and Protasov, Vladimir and Blondel,
                  Vincent D.",
 authauthor    = "Jungers, Rapha{\"e}l and Protasov, Vladimir and Blondel,
                  Vincent",
 title         = "Efficient algorithms for deciding the type of growth of
                  products of integer matrices",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2008",
 volume        = "428",
 number        = "10",
 pages         = "2296--2311",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15A36 (05C90 15A42)",
 mrnumber      = "2405246 (2009a:15065)",
 mrreviewer    = "Enric Ventura Capell",
 zblnumber     = "1145.65030",
 zblreviewer   = "R{\'e}mi Vaillancourt (Ottawa)",
 doi           = "10.1016/j.laa.2007.08.001",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379507003436",
 annote        = "For a given set $\Sigma$ of matrices, the joint spectral
                  radius of $\Sigma$, denoted $\rho(\Sigma)$, is defined by
                  the limit $\rho(\Sigma)=\lim_{t\rightarrow
                  \infty}\max\{\|A_1\cdots A_t\|^{1/t}: A_t\in\Sigma\}$
                  independently of any norm. For any finite set $\Sigma$ of
                  $n\times n$ matrices with nonnegative integer entries, the
                  authors show that there is a polynomial algorithm that
                  decides between the four cases: $\rho=0$, $\rho=1$ and
                  bounded, $\rho=1$ and polynomial growth, and $\rho>1$. The
                  polynomial solvability for $\rho>1$ is somewhat
                  surprising.",
}

@ARTICLE{JunProt:SIAMJSC11,
 author        = "Jungers, Rapha{\"e}l M. and Protasov, Vladimir
                  {\relax{}Yu}.",
 authauthor    = "Jungers, Rapha{\"e}l and Protasov, Vladimir",
 title         = "Fast methods for computing the {$p$}-radius of matrices",
 journal       = "SIAM J. Sci. Comput.",
 fjournal      = "SIAM Journal on Scientific Computing",
 year          = "2011",
 volume        = "33",
 number        = "3",
 pages         = "1246--1266",
 coden         = "SJOCE3",
 issn          = "1064-8275",
 mrclass       = "15A60 (90C22)",
 mrnumber      = "2813238 (2012f:15034)",
 mrreviewer    = "Jiu Ding",
 zblnumber     = "1236.65036",
 zblreviewer   = "Constantin Popa (Constan{\c{t}}a)",
 doi           = "10.1137/090777906",
 url           = "https://epubs.siam.org/doi/10.1137/090777906",
 annote        = "The $p$-radius characterizes the average rate of growth of
                  norms of matrices in a multiplicative semigroup. This
                  quantity has found several applications in recent years.
                  The paper is concerned with the numerical approximation of
                  the $p$-radius of a set of matrices. The authors prove that
                  the complexity of its approximation increases exponentially
                  with $p$ and propose a series of approximations that
                  converge to the $p$-radius with a priori computable
                  accuracy.",
}

@INCOLLECTION{JPB:LNCS08,
 author        = "Jungers, Rapha{\"e}l M. and Protasov, Vladimir {\relax{}Yu}.
                  and Blondel, Vincent D.",
 authauthor    = "Jungers, Rapha{\"e}l and Protasov, Vladimir and Blondel,
                  Vincent",
 title         = "Computing the Growth of the Number of Overlap-Free Words
                  with Spectra of Matrices",
 booktitle     = "{LATIN} 2008: {T}heoretical Informatics (Proceedings of the
                  8th Latin American Symposium, B{\'u}zios, Brazil, April
                  7--11, 2008)",
 series        = "Lecture Notes in Computer Science",
 publisher     = "Springer Berlin Heidelberg",
 year          = "2008",
 volume        = "4957",
 pages         = "84--93",
 mrclass       = "68R15 (05A16)",
 mrnumber      = "2472727",
 zblnumber     = "1136.68471",
 doi           = "10.1007/978-3-540-78773-0_8",
 url           = "https://link.springer.com/chapter/10.1007/978-3-540-78773-0_8",
 annote        = "Overlap-free words are words over the alphabet $A=\{a, b\}$
                  that do not contain factors of the form $xvxvx$, where
                  $x\in A$ and $v\in A^*$. We analyze the asymptotic growth
                  of the number $u_n$ of overlap-free words of length $n$. We
                  obtain explicit formulas for the minimal and maximal rates
                  of growth of $u_n$ in terms of spectral characteristics
                  (the lower spectral radius and the joint spectral radius)
                  of one set of matrices of dimension $20$. Using these
                  descriptions we provide estimates of the rates of growth
                  that are within $0.4\%$ and $0.03\%$ of their exact value.
                  The best previously known bounds were within $11\%$ and
                  $3\%$ respectively. We prove that $u_n$ actually has the
                  same growth for ``almost all'' $n$. This ``average'' growth
                  is distinct from the maximal and minimal rates and can also
                  be expressed in terms of a spectral quantity (the Lyapunov
                  exponent). We use this expression to estimate it.",
}

@ARTICLE{JPB:TCM09,
 author        = "Jungers, Rapha{\"e}l M. and Protasov, Vladimir {\relax{}Yu}.
                  and Blondel, Vincent D.",
 authauthor    = "Jungers, Rapha{\"e}l and Protasov, Vladimir and Blondel,
                  Vincent",
 title         = "Overlap-free words and spectra of matrices",
 journal       = "Theoret. Comput. Sci.",
 fjournal      = "Theoretical Computer Science",
 year          = "2009",
 volume        = "410",
 number        = "38-40",
 pages         = "3670--3684",
 eprinttype    = "arXiv",
 eprint        = "0709.1794",
 coden         = "TCSDI",
 issn          = "0304-3975",
 mrclass       = "68R15 (05A05 15A18)",
 mrnumber      = "2553320 (2010k:68081)",
 zblnumber     = "1171.68035",
 doi           = "10.1016/j.tcs.2009.04.022",
 url           = "https://www.sciencedirect.com/science/article/pii/S0304397509003387",
 annote        = "Overlap-free words are words over the binary alphabet
                  $A=\{a,b\}$ that do not contain factors of the form
                  $xvxvx$, where $x\in A$ and $v\in A^{*}$. We analyze the
                  asymptotic growth of the number $u_n$ of overlap-free words
                  of length $n$ as $n\rightarrow \infty $. We obtain explicit
                  formulas for the minimal and maximal rates of growth of
                  $u_n$ in terms of spectral characteristics (the joint
                  spectral subradius and the joint spectral radius) of
                  certain sets of matrices of dimension $20\times 20$. Using
                  these descriptions we provide new estimates of the rates of
                  growth that are within $0.4\%$ and $0.03\%$ of their exact
                  values. The best previously known bounds were within $11\%$
                  and $3\%$, respectively. We then prove that the value of
                  $un$ actually has the same rate of growth for ``almost
                  all'' natural numbers $n$. This average growth is distinct
                  from the maximal and minimal rates and can also be
                  expressed in terms of a spectral quantity (the Lyapunov
                  exponent). We use this expression to estimate it. In order
                  to obtain our estimates, we introduce new algorithms to
                  compute the spectral characteristics of sets of matrices.
                  These algorithms can be used in other contexts and are of
                  independent interest.",
}

@BOOK{KasBh:2000,
 author        = "Kaszkurewicz, Eugenius and Bhaya, Amit",
 title         = "Matrix diagonal stability in systems and computation",
 publisher     = "Birkh{\"a}user Boston Inc.",
 address       = "Boston, MA",
 year          = "2000",
 pages         = "xiv+267",
 isbn          = "0-8176-4088-6",
 mrclass       = "34D99 (34D20 39A10 65F30 93D05)",
 mrnumber      = "1733604 (2001c:34109)",
 mrreviewer    = "Norihiko Adachi",
 zblnumber     = "0951.93058",
 zblreviewer   = "Christina Diakaki (Chania)",
 doi           = "10.1007/978-1-4612-1346-8",
 url           = "https://link.springer.com/book/10.1007/978-1-4612-1346-8/page/1",
 annote        = "The book presents a collection of results, observations, and
                  examples related to dynamical systems described by linear
                  and nonlinear ordinary differential and difference
                  equations. In particular, it considers dynamical systems
                  that are susceptible to analysis by the Lyapunov approach.
                  The book consists of 6 chapters. The first chapter is
                  devoted to examples that originate from different
                  applications and illustrate the way in which some special
                  classes of dynamical systems dovetail with the concepts of
                  matrix diagonal stability and the associated diagonal-type
                  Lyapunov functions. The second chapter presents the matrix
                  theory concepts of diagonal stability and also of
                  $D$-stability and gives the properties of these classes of
                  matrices. Chapter 3 introduces classes of dynamical systems
                  that admit diagonal-type Lyapunov functions and gives the
                  basic stability results. Chapter 4 shows how Jacobi-type
                  iterative methods for the solution of linear and nonlinear
                  equations and, more specifically, asynchronous versions of
                  these methods can be analysed, based on diagonal-type
                  Lyapunov functions. Chapter 5 shows and discusses the
                  occurrence of the diagonal structure, introduced in the
                  first chapter, in several classes of dynamical systems that
                  include neural networks, digital filters, passive PLC
                  circuits, and ecosystems. Finally, chapter 6 discusses
                  various applications of dynamical systems in which
                  diagonally stable structures can be used
                  advantageously.\par The book provides an essential
                  reference for new methods and analysis related to dynamical
                  systems described by linear and nonlinear ordinary
                  differential equations and difference equations. As such,
                  it is addressed to researchers, professionals, and
                  graduates in applied mathematics, control engineering,
                  stability of dynamical systems, convergence of algorithms,
                  and scientific computation. Familiarity with linear algebra
                  and matrix theory, as well as difference and differential
                  equations, is the mathematical background expected from its
                  readers. A prior knowledge of systems and control theory,
                  including Lyapunov stability theory, is also expected, at
                  least in sufficient measure to provide motivation for the
                  problems studied.",
}

@INCOLLECTION{KisShulTur:20,
 author        = "Kissin, Edward and Shulman, Victor S. and Turovskii,
                  {\relax{}Yu}ri{\u\i} V.",
 authauthor    = "Kissin, Edward and Shulman, Victor and Turovskii,
                  {\relax{}Yu}ri{\u\i}",
 title         = "From {L}omonosov {L}emma to radical approach in joint
                  spectral radius theory",
 booktitle     = "The mathematical legacy of {V}ictor {L}omonosov---operator
                  theory",
 series        = "Adv. Anal. Geom.",
 publisher     = "De Gruyter, Berlin",
 year          = "2020",
 volume        = "2",
 pages         = "205--230",
 eprinttype    = "arXiv",
 eprint        = "2005.02743",
 mrclass       = "47A15 (47L10)",
 mrnumber      = "4312040",
 zblnumber     = "07279804",
 doi           = "10.1515/9783110656756-015",
 url           = "https://www.degruyter.com/view/book/9783110656756/10.1515/9783110656756-015.xml",
 annote        = "In this paper we discuss the infinite-dimensional
                  generalizations of the famous theorem of Berger--Wang
                  (generalized Berger--Wang formulas) and give an
                  operator-theoretic proof of I. Morris's theorem about
                  coincidence of three essential joint spectral radius,
                  related to these formulas. Further we develop
                  Banach-algebraic approach based on the theory of
                  topological radicals, and obtain some new results about
                  these radicals.",
}

@ARTICLE{KisShulTur:JMS22,
 author        = "Kissin, Edward and Turovskii, {\relax{}Yu}ri{\u\i} V. and
                  Shulman, Victor S.",
 authauthor    = "Kissin, Edward and Shulman, Victor and Turovskii,
                  {\relax{}Yu}ri{\u\i}",
 title         = "On the theory of topological radicals",
 year          = "2022",
 journal       = "J. Math. Sci.",
 fjournal      = "Journal of Mathematical Sciences",
 volume        = "263",
 number        = "6",
 pages         = "805--859",
 mrclass       = "46H05 (16N80 17B60 46H35)",
 mrnumber      = "3914375",
 mrreviewer    = "Mart Abel",
 zblnumber     = "07552500",
 doi           = "10.1007/s10958-022-05969-8",
 url           = "https://link.springer.com/article/10.1007/s10958-022-05969-8",
 annote        = "In this paper, we review main directions and results of the
                  theory of topological radicals. We consider applications to
                  different problems in the theory of operators and Banach
                  algebras.",
}

@ARTICLE{KlamCzorNiez:BPASTS13,
 author        = "Klamka, Jerzy and Czornik, Adam and Niezabitowski,
                  Micha{\l}",
 title         = "Stability and controllability of switched linear dynamical
                  systems",
 journal       = "Bull. Pol. Acad. Sci. Tech. Sci.",
 year          = "2013",
 volume        = "61",
 number        = "3",
 pages         = "547--555",
 month         = sep,
 doi           = "10.2478/bpasts-2013-0055",
 url           = "http://journals.pan.pl/dlibra/publication/97158/edition/83745/content",
 annote        = "The study of properties of switched and hybrid systems gives
                  rise to a number of interesting and challenging
                  mathematical problems. This paper aims to briefly survey
                  recent results on stability and controllability of switched
                  linear systems. First, the stability analysis for switched
                  systems is reviewed. We focus on the stability analysis for
                  switched linear systems under arbitrary switching, and we
                  highlight necessary and sufficient conditions for
                  asymptotic stability. After that, we review the
                  controllability results.",
}

@ARTICLE{Klep:AIT85:e,
 author        = "Kleptsyn, A. F.",
 authauthor    = "Kleptsyn, A.",
 title         = "Stability of desynchronized complex systems of a special
                  type",
 journal       = "Avtomat. i Telemekh.",
 fjournal      = "Akademiya Nauk SSSR. Avtomatika i Telemekhanika",
 year          = "1985",
 number        = "4",
 pages         = "169--171",
 coden         = "AVTEAI",
 issn          = "0005-2310",
 mrclass       = "93D05",
 mrnumber      = "807476 (86j:93025)",
}

@ARTICLE{KKKK:AiT83:7:e,
 author        = "Kleptsyn, A. F. and Kozyakin, V. S. and Krasnosel'skij, M.
                  A. and Kuznetsov, N. A.",
 authauthor    = "Kleptsyn, A. and Kozyakin, Victor and Krasnosel'skii, Mark
                  and Kuznetsov, N.",
 title         = "Effect of small synchronization errors on stability of
                  complex systems. {I}",
 journal       = "Autom. Remote Control",
 fjournal      = "Automation and Remote Control",
 year          = "1983",
 volume        = "44",
 number        = "7",
 pages         = "861--867",
 issn          = "0005-1179",
 zblnumber     = "0539.93067",
 zblreviewer   = "J.Gayek",
 language      = "english",
 annote        = "A necessary and sufficient condition is given for the
                  asymptotic stability of the origin for $n$-dimensional
                  autonomous discrete linear systems with small phase lags in
                  switching. Furthermore, a sufficient condition for the
                  origin to be absolutely stable with respect to phase errors
                  is presented.",
}

@ARTICLE{KKKK:DAN84:e,
 author        = "Kleptsyn, A. F. and Kozyakin, V. S. and Krasnosel'skij, M.
                  A. and Kuznetsov, N. A.",
 authauthor    = "Kleptsyn, A. and Kozyakin, Victor and Krasnosel'skii, Mark
                  and Kuznetsov, N.",
 title         = "Stability of desynchronized systems",
 journal       = "Sov. Phys., Dokl.",
 fjournal      = "Soviet Physics. Doklady",
 year          = "1984",
 volume        = "29",
 pages         = "92--94",
 issn          = "0038-5689",
 mrclass       = "93D20",
 mrnumber      = "734942 (86e:93085)",
 zblnumber     = "0593.93046",
 zblreviewer   = "G. Leonov",
 language      = "english",
 annote        = "The concept of phase and frequency desynchronized systems is
                  introduced. These systems arise in cases when their state
                  variables are switched independently (by using independent
                  functional devices, desynchronized elements such as
                  extrapolators, memory elements etc.). For such systems
                  asymptotic stability theorems are produced.",
}

@ARTICLE{KKKK:AiT84:3:e,
 author        = "Kleptsyn, A. F. and Kozyakin, V. S. and Krasnosel'skij, M.
                  A. and Kuznetsov, N. A.",
 authauthor    = "Kleptsyn, A. and Kozyakin, Victor and Krasnosel'skii, Mark
                  and Kuznetsov, N.",
 title         = "Effect of small synchronization errors on stability of
                  complex systems. {II}",
 journal       = "Autom. Remote Control",
 fjournal      = "Automation and Remote Control",
 year          = "1984",
 volume        = "45",
 number        = "3",
 pages         = "309--314",
 issn          = "0005-1179",
 zblnumber     = "0618.93057",
 language      = "english",
 annote        = "[For part I see~\cite{KKKK:AiT83:7:e}]. Classes of linear
                  systems with digital elements are considered that are
                  insensitive to synchronization errors.",
}

@ARTICLE{KKKK:AiT84:8:e,
 author        = "Kleptsyn, A. F. and Kozyakin, V. S. and Krasnosel'skij, M.
                  A. and Kuznetsov, N. A.",
 authauthor    = "Kleptsyn, A. and Kozyakin, Victor and Krasnosel'skii, Mark
                  and Kuznetsov, N.",
 title         = "Effect of small synchronization errors on stability of
                  complex systems. {III}",
 journal       = "Autom. Remote Control",
 fjournal      = "Automation and Remote Control",
 year          = "1984",
 volume        = "45",
 number        = "8",
 pages         = "1014--1018",
 issn          = "0005-1179",
 zblnumber     = "0604.93047",
 zblreviewer   = "J.Gayek",
 language      = "english",
 annote        = "[For the previous parts
                  see~\cite{KKKK:AiT83:7:e,KKKK:AiT84:3:e}]. \par Sufficiency
                  criteria are presented for the asymptotic stability of the
                  origin for systems governed by $n$-dimensional autonomous
                  discrete-time linear equations with desynchronized
                  frequency of switching amongst the state components.",
}

@ARTICLE{KKKK:MCS84,
 author        = "Kleptsyn, A. F. and Krasnosel'ski{\u\i}, M. A. and
                  Kuznetsov, N. A. and Kozjakin, V. S.",
 authauthor    = "Kleptsyn, A. and Krasnosel'skii, Mark and Kuznetsov, N. and
                  Kozyakin, Victor",
 title         = "Desynchronization of linear systems",
 journal       = "Math. Comput. Simulation",
 fjournal      = "Mathematics and Computers in Simulation",
 year          = "1984",
 volume        = "26",
 number        = "5",
 pages         = "423--431",
 coden         = "MCSIDR",
 issn          = "0378-4754",
 mrclass       = "93D05 (93C55)",
 mrnumber      = "762920 (85j:93041)",
 zblnumber     = "0552.93051",
 zblreviewer   = "H. Fischer",
 doi           = "10.1016/0378-4754(84)90106-X",
 url           = "https://www.sciencedirect.com/science/article/pii/037847548490106X",
 annote        = "The studies of dynamics inherent in control systems which
                  incorporate sampled data elements such as extrapolators,
                  keys, memory elements, etc. are usually reduced to the
                  analysis of difference equations. However, this approach
                  tends to disregard the inevitable small desynchronization
                  of times when sampled elements are connected. In some
                  systems this desynchronization does not influence the
                  stability; otheres may be destabilized by any infinitesimal
                  desynchronization; finally, some unstable synchronized
                  systems can become asymptotically stable following the
                  introduction of any infinitesimal desynchronization. These
                  facts account for many phenomena observed in the
                  engineering practice that looked enigmatic when the
                  mathematical models ignored small desynchronization. In
                  designing the controllers the stability can be achieved by
                  introduction of lags into the system.\par The paper
                  investigates linear discrete-time (sampled-data) systems
                  with respect to stability. The systems are assumed to be
                  desynchronized in time, i.e. the time instants for sampling
                  the different state variables are incommensurable. Two
                  types of desynchronization are distinguished: a phase one
                  and a frequency one. Whereas phase-desynchronization
                  describes sampling instants $t_i$ to be $t_ i=nh+\tau_i$,
                  frequency-desynchronization describes them as
                  $t_i=n(h+\delta_i)$. Various theorems for determining the
                  stability of such systems are given without proofs, which
                  will be published in another paper. It is appreciable that
                  six examples illustrate the application of the results.
                  Here it turns out, that a system may be stable if it is
                  synchronized and may become unstable if small frequency
                  shifts are introduced: If the frequency shifts become
                  larger and larger, it is interesting to observe stability
                  for the same system again.",
}

@INPROCEEDINGS{Koz:CDC05:e,
 author        = "Kozyakin, V.",
 title         = "A Dynamical Systems Construction of a Counterexample to the
                  Finiteness Conjecture",
 booktitle     = "Proceedings of the 44th IEEE Conference on Decision and
                  Control, 2005 and 2005 European Control Conference.
                  CDC-ECC'05",
 year          = "2005",
 pages         = "2338--2343",
 doi           = "10.1109/CDC.2005.1582511",
 url           = "https://ieeexplore.ieee.org/document/1582511",
 annote        = "In 1995 \emph{J.\,C.~Lagarias} and
                  \emph{Y.~Wang}~\cite{LagWang:LAA95} conjectured that the
                  generalized spectral radius of a finite set of matrices can
                  be attained on a finite product of matrices. The first
                  counterexample to this Finiteness Conjecture was given in
                  2002 by \emph{T.~Bousch} and
                  \emph{J.~Mairesse}~\cite{BM:JAMS02}. In 2003
                  \emph{V.\,D.~Blondel}, \emph{J.~Theys} and
                  \emph{A.\,A.~Vladimirov}~\cite{BTV:SIAMJMA03} proposed
                  another proof of a counterexample to the Finiteness
                  Conjecture which extensively exploited combinatorial
                  properties of matrix products. In the paper, it is proposed
                  one more proof of a counterexample of the Finiteness
                  Conjecture fulfilled in a traditional manner of the theory
                  of dynamical systems. It is presented description of the
                  structure of trajectories with the maximal growing rate in
                  terms of extremal norms and associated with them so-called
                  extremal trajectories. The construction of the
                  counterexample is based on a detailed analysis of
                  properties of extremal norms of two-dimensional positive
                  matrices in which the technique of the Gram symbols is
                  essentially used. At last, notions and properties of the
                  rotation number for discontinuous orientation preserving
                  circle maps play significant role in the proof.",
}

@ARTICLE{Koz:AiT90:10:e,
 author        = "Kozyakin, V. S.",
 authauthor    = "Kozyakin, Victor",
 title         = "Absolute stability of systems with asynchronous sampled-data
                  elements",
 journal       = "Autom. Remote Control",
 fjournal      = "Automation and Remote Control",
 year          = "1990",
 volume        = "51",
 number        = "10",
 pages         = "1349--1355",
 issn          = "0005-1179",
 mrclass       = "93D05 (93C57)",
 mrnumber      = "1088520 (91j:93070)",
 zblnumber     = "0737.93055",
 zblreviewer   = "E. Yaz (Fayetteville)",
 language      = "english",
 annote        = "The author considers the problem of asynchronous operation
                  of elements in sampled-data systems. The concepts of
                  absolute neutral and asymptotic stability with respect to
                  different asynchronous operation classes are introduced and
                  results characterizing such properties of asynchronous
                  systems are presented.",
}

@ARTICLE{Koz:AiT90:6:e,
 author        = "Kozyakin, V. S.",
 authauthor    = "Kozyakin, Victor",
 title         = "Algebraic unsolvability of problem of absolute stability of
                  desynchronized systems",
 journal       = "Autom. Remote Control",
 fjournal      = "Automation and Remote Control",
 year          = "1990",
 volume        = "51",
 number        = "6",
 pages         = "754--759",
 issn          = "0005-1179",
 mrclass       = "93D09 (93C55)",
 mrnumber      = "1071607 (91e:93065)",
 zblnumber     = "0737.93056",
 language      = "english",
 annote        = "In complex control systems containing sampled-data elements,
                  it is possible that these elements operate asynchronously.
                  In some cases asynchronous character of operation of
                  sampled-data elements does not influence stability of
                  system. In other cases any small desynchronization of the
                  updating moments of sampled-data elements leads to dramatic
                  changes of dynamics of a control system, and the system
                  loses stability. Last years there is begun intensive
                  studying of the effects connected with asynchronous
                  operation of control systems; both necessary, and
                  sufficient stability conditions for various classes of
                  asynchronous systems were obtained. At the same time no one
                  succeed in finding general, effectively verified criteria
                  of stability of asynchronous systems, similar to known for
                  synchronous systems. The problem on stability of linear
                  asynchronous systems has appeared more difficult than the
                  problem on stability of synchronous systems. In the paper,
                  attempt of formal explanation of complexity of the
                  stability analysis problem for linear asynchronous systems
                  is undertaken. It is shown that there are no criteria of
                  absolute stability of linear asynchronous systems
                  consisting of a finite number of arithmetic operations.",
}

@ARTICLE{Koz:AiT90:8:e,
 author        = "Kozyakin, V. S.",
 authauthor    = "Kozyakin, Victor",
 title         = "Stability of phase-frequency desynchronized systems under a
                  perturbation of the switching instants of the components",
 journal       = "Autom. Remote Control",
 fjournal      = "Automation and Remote Control",
 year          = "1990",
 volume        = "51",
 number        = "8",
 pages         = "1034--1040",
 issn          = "0005-1179",
 mrclass       = "93D09",
 mrnumber      = "1080599 (91j:93078)",
 zblnumber     = "0735.93067",
 zblreviewer   = "D. Franke (Hamburg)",
 language      = "english",
 annote        = "The paper studies a control problem for systems with
                  discrete elements which has found little attention in the
                  past. Two distance subsystems are assumed to exchange
                  information periodically. This is done non-synchronously,
                  because such subsystem is assumed to have its individual
                  periodicity. Asymptotic stability of the zero equilibrium
                  position of the overall system is investigated in presence
                  of small perturbations of the periods and the phases of
                  switching of the subsystems.",
}

@ARTICLE{Koz:DAN90:311:e,
 author        = "Kozyakin, V. S.",
 authauthor    = "Kozyakin, Victor",
 title         = "Stability analysis of asynchronous systems by methods of
                  symbolic dynamics",
 journal       = "Sov. Phys., Dokl.",
 fjournal      = "Soviet Physics. Doklady",
 year          = "1990",
 volume        = "35",
 number        = "3",
 pages         = "218--220",
 issn          = "0038-5689",
 mrclass       = "93C57 (54H20)",
 mrnumber      = "1075682 (92c:93056)",
 mrreviewer    = "Nikita E. Barabanov",
 zblnumber     = "0713.93045",
 annote        = "The dynamics of autonomous control systems with sampled-data
                  elements is described, by the equations \[x_i(T^{ij}+0)=
                  f_i[x_1(T^{ij}-0),\ldots,x_k(T^{ij}-0)],\quad
                  i=1,\ldots,k,\] where each vector function $x_i(t)$ with
                  values in $R_{n_i}$ is constant on the intervals
                  $T^{ij}<t<T^{ij+1}$. In the classical theory of
                  sampled-data systems, the switching times $T^{ij}$ are the
                  same for all components $x_i$, i.e.,
                  $T^{1j}=T^{2j}=\dots=T^{kj}$. A rich arsenal of methods for
                  the analysis of the dynamics of the systems under
                  consideration has been developed for this case.\par An
                  important class are phase-frequency asynchronous systems
                  with switching times $T^{ij}=\phi_i+j \omega_i$.
                  \emph{A.\,F.~Kleptsyn}~\cite{Klep:AIT85:e} and
                  \emph{A.\,F.~Kleptsyn}, \emph{M.\,A.~Krasnosel'skij},
                  \emph{N.\,A.~Kuznetsov} and
                  \emph{V.\,S.~Kozyakin}~\cite{KKKK:MCS84} noted that the
                  stability of these systems is closely linked with the
                  properties of sequences of symbols called shift texts. This
                  observation has led to important theorems on stability of
                  linear two-component phase-frequency asynchronous control
                  systems.",
}

@ARTICLE{Koz:AiT91:e,
 author        = "Kozyakin, V. S.",
 authauthor    = "Kozyakin, Victor",
 title         = "Stability of linear desynchronized systems with unsymmetric
                  matrices",
 journal       = "Autom. Remote Control",
 fjournal      = "Automation and Remote Control",
 year          = "1991",
 volume        = "52",
 number        = "7",
 pages         = "928--933",
 issn          = "0005-1179",
 mrclass       = "93D09",
 mrnumber      = "1139401 (92h:93064)",
 zblnumber     = "0744.93083",
 language      = "english",
 annote        = "A new criterion of asymptotic stability is established for
                  linear desynchronized systems. We show that the problem of
                  preserving absolute asymptotic stability in the presence of
                  small matrix perturbation in linear desynchronized systems
                  is equivalent to the problem of stability in the presence
                  of persistent perturbations.",
}

@ARTICLE{Koz:DAN91:e,
 author        = "Kozyakin, V. S.",
 authauthor    = "Kozyakin, Victor",
 title         = "Perturbation of linear asynchronous systems",
 journal       = "Sov. Phys., Dokl.",
 fjournal      = "Soviet Physics. Doklady",
 year          = "1991",
 volume        = "36",
 number        = "1",
 pages         = "16--17",
 issn          = "0038-5689",
 mrclass       = "93D09",
 mrnumber      = "1102773",
 zblnumber     = "0752.93064",
 zblreviewer   = "L. Rodman (Williamsburg)",
 annote        = "A system $W$ is described by its state space
                  $X=\{x=(x_1,x_2,\ldots,x_N)^T: x_i\in\mathbb{R}^{n_i}\}$
                  and dynamics $x(0)=0$, $x(n+1)=A_{w(n)}x(n)+F(n)$, $F(n)\in
                  X_{w(n)}$, $n=0,1,\ldots$, where $w=\{w(n)\}^\infty_{n=1}$
                  is a sequence of subsets $w(n)\subseteq\{1,\ldots,N\}$.
                  Here $A=[a_{ij}]^N_{i,j=1}$ is a fixed block matrix,
                  $A_{w(n)}$ is obtained from $A$ by replacing its block rows
                  corresponding to indices $i\not\in w(n)$ with the
                  corresponding rows of the identity matrix, and
                  \[X_{w(n)}=\{(x_1,\ldots,x_n)^T\in X:\quad
                  x_i=0\quad\text{if}\quad i\not\in w(n)\}.\] The system $W$
                  with the matrix $A$ is called Perron absolutely stable if
                  there is $\beta<\infty$ such that for any sequence $w$ of
                  subsets in $\{1,\ldots,N\}$ and any choice of $F(n)\in
                  X_{w(n)}$ satisfying $\|F(n)\|\le 1$, the state $x(n)$
                  satisfies the inequality $\|x(n)\|\le\beta$ for all $n$.
                  Perron absolutely stability is characterized in terms of
                  existence of a norm in $X$ with special properties.",
}

@ARTICLE{Koz:AiT03:9:e,
 author        = "Kozyakin, V. S.",
 authauthor    = "Kozyakin, Victor",
 title         = "Indefinability in o-minimal structures of finite sets of
                  matrices whose infinite products converge and are bounded
                  or unbounded",
 journal       = "Autom. Remote Control",
 fjournal      = "Automation and Remote Control",
 year          = "2003",
 volume        = "64",
 number        = "9",
 pages         = "1386--1400",
 issn          = "0005-1179",
 mrclass       = "03C64 (14P10 93B27)",
 mrnumber      = "2090805 (2005h:03070)",
 mrreviewer    = "S. R. Kogalovski{\u\i}",
 zblnumber     = "1078.93017",
 doi           = "10.1023/A:1026091717271",
 url           = "https://link.springer.com/article/10.1023/A:1026091717271",
 language      = "english",
 annote        = "This paper is concerned with the convergence and boundedness
                  or unboundedness of the set of all possible matrix products
                  with coefficients belonging to some finite set, i.e., the
                  problem to which many problems of control theory and
                  mathematics are reduced. The indefinability of this problem
                  in o-minimal structures containing semialgebraic sets,
                  which can be regarded as a characteristic for the
                  complexity of the problem, is demonstrated. The result
                  shows, in particular, that the solution of our problem
                  cannot be found as a finite Boolean combination of
                  conditions containing a finite number of ordinary
                  arithmetical operations of addition, subtraction, and
                  multiplication, as well as exponentiation and application
                  of bounded analytic functions.",
}

@TECHREPORT{Koz:BCRI03-13,
 author        = "Kozyakin, V. S.",
 authauthor    = "Kozyakin, V.",
 title         = "Asynchronous Systems: A Short Survey and Problems",
 institution   = "Boole Centre for Research in Informatics, University College
                  Cork --- National University of Ireland",
 address       = "Cork",
 year          = "2003",
 number        = "13/2003",
 pagetotal     = "19",
 type          = "Preprint",
 month         = may,
 url           = "https://www.researchgate.net/publication/228863199_Asynchronous_Systems_A_Short_Survey_and_Problems",
 annote        = "Looking at the dynamics of the system described by the
                  equation $x(n + 1) = f(x(n))$ one may say that coordinates
                  of the vector $x = {x_{1}, x_{2},\ldots , x_{N}}$ are
                  updated synchronously. What happens with the system if
                  coordinates of the vector $x$ are updated asynchronously,
                  i.e., if at a given moment $n$ only coordinates with
                  indices $i$ from some set $w(n) \subseteq {1,2,\ldots,N}$
                  are changed in accordance with the law $x_{i}(n+1) =
                  f_{i}(x(n))$ while others remain intact? This is the main
                  topic which is discussed in the paper.",
}

@TECHREPORT{Koz:WIAS05,
 author        = "Kozyakin, V. S.",
 authauthor    = "Kozyakin, V.",
 title         = "Proof of a Counterexample to the Finiteness Conjecture in
                  the Spirit of the Theory of Dynamical Systems",
 institution   = "Weierstra{\ss}-Institut f{\"u}r Angewandte Analysis und
                  Stochastik",
 address       = "Berlin",
 year          = "2005",
 number        = "1005",
 pagetotal     = "46",
 type          = "Preprint",
 month         = jan,
 url           = "https://www.researchgate.net/publication/234109568_Proof_of_a_Counterexample_to_the_Finiteness_Conjecture_in_the_Spirit_of_the_Theory_of_Dynamical_Systems",
 annote        = "In 1995 \emph{J.\,C.~Lagarias} and
                  \emph{Y.~Wang}~\cite{LagWang:LAA95} conjectured that the
                  generalized spectral radius of a finite set of square
                  matrices can be attained on a finite product of matrices.
                  The first counterexample to this Finiteness Conjecture was
                  given in 2002 by \emph{T.~Bousch} and
                  \emph{J.~Mairesse}~\cite{BM:JAMS02} and their proof was
                  based on measure-theoretical ideas. In 2003
                  \emph{V.\,D.~Blondel}, \emph{J.~Theys} and
                  \emph{A.\,A.~Vladimirov}~\cite{BTV:SIAMJMA03} proposed
                  another proof of a counterexample to the Finiteness
                  Conjecture which extensively exploited combinatorial
                  properties of permutations of products of positive
                  matrices. In the paper, it is proposed one more proof of a
                  counterexample of the Finiteness Conjecture fulfilled in a
                  rather traditional manner of the theory of dynamical
                  systems. It is presented description of the structure of
                  trajectories with the maximal growing rate in terms of
                  extremal norms and associated with them so called extremal
                  trajectories. The construction of the counterexample is
                  based on a detailed analysis of properties of extremal
                  norms of two-dimensional positive matrices in which the
                  technique of the Gram symbols is essentially used. At last,
                  notions and properties of the rotation number for
                  discontinuous orientation preserving circle maps play
                  significant role in the proof.",
}

@ARTICLE{Koz:INFOPROC06:e,
 author        = "Kozyakin, V. S.",
 authauthor    = "Kozyakin, V.",
 title         = "Structure of extremal trajectories of discrete linear
                  systems and the finiteness conjecture",
 journal       = "Autom. Remote Control",
 fjournal      = "Automation and Remote Control",
 year          = "2007",
 volume        = "68",
 number        = "1",
 pages         = "174--209",
 issn          = "0005-1179",
 zblnumber     = "1195.93082",
 doi           = "10.1134/S0005117906040171",
 url           = "https://link.springer.com/article/10.1134/S0005117906040171",
 annote        = "\emph{J.\,C.~Lagarias} and
                  \emph{Y.~Wang}~\cite{LagWang:LAA95} conjectured that the
                  generalized spectral radius of a finite set of square
                  matrices can be attained on a finite product of matrices.
                  The first counterexample to this Finiteness Conjecture was
                  given by \emph{T.~Bousch} and
                  \emph{J.~Mairesse}~\cite{BM:JAMS02} and their proof was
                  based on measure-theoretical ideas. \emph{V.\,D.~Blondel},
                  \emph{J.~Theys} and
                  \emph{A.\,A.~Vladimirov}~\cite{BTV:SIAMJMA03} proposed
                  another proof of a counterexample to the Finiteness
                  Conjecture which extensively exploited combinatorial
                  properties of permutations of products of positive
                  matrices. In the control theory, so as in the general
                  theory of dynamical systems, the notion of generalized
                  spectral radius is used basically to describe the rate of
                  growth or decrease of the trajectories generated by matrix
                  products. In this context, the above mentioned methods are
                  not enough satisfactory (from the point of view of the
                  author, of course) since they give no description of the
                  structure of the trajectories with the maximal growing rate
                  (or minimal decreasing rate). In connection with this, in
                  2005 \emph{V.\,S.~Kozyakin}~\cite{Koz:WIAS05} presented one
                  more proof of the counterexample to the Finiteness
                  Conjecture fulfilled in the spirit of the theory of
                  dynamical systems. Unfortunately, the developed approach
                  did not cover the class of matrices considered by Blondel,
                  Theys and Vladimirov. The goal of the present paper is to
                  compensate for this deficiency in the previous approach.",
}

@ARTICLE{Koz:DAN09:e,
 author        = "Kozyakin, V. S.",
 authauthor    = "Kozyakin, Victor",
 title         = "On the computational aspects of the theory of joint spectral
                  radius",
 journal       = "Dokl. Math.",
 fjournal      = "Doklady Mathematics",
 year          = "2009",
 volume        = "80",
 number        = "1",
 pages         = "487--491",
 issn          = "1064-5624",
 mrclass       = "15A18 (15A60 47A10)",
 mrnumber      = "2573049 (2010k:15023)",
 mrreviewer    = "Adolf Rhodius",
 zblnumber     = "1190.93035",
 doi           = "10.1134/S1064562409040097",
 url           = "https://link.springer.com/article/10.1134/S1064562409040097",
 annote        = "One of the most prominent tool to compute the the joint
                  spectral radius of a matrix set is the so-called
                  generalized Gelfand formula which represents the joint
                  spectral radius as a limit of the weighted norms of matrix
                  products with factors taken from the given matrix set.
                  Unfortunately, the range of applicability of this formula
                  is substantially restricted by a lack of estimates for the
                  rate of convergence to the corresponding limit. In the
                  paper this deficiency is made up to some extent. We
                  establish explicit computable estimates of the joint
                  spectral radius by the norms of matrix products with
                  factors taken from the given matrix set, and propose two
                  computational iterative relaxation procedures to
                  approximate the joint spectral radius of matrix sets.",
}

@MISC{Koz:ArXiv13,
 author        = "Kozyakin, V. S.",
 authauthor    = "Kozyakin, V.",
 title         = "Algebraic Unsolvability of Problem of Absolute Stability of
                  Desynchronized Systems Revisited",
 howpublished  = "ArXiv.org e-Print archive",
 year          = "2013",
 month         = jan,
 eprinttype    = "arXiv",
 eprint        = "1301.5409",
 doi           = "10.48550/arXiv.1301.5409",
 annote        = "In the author's article~\cite{Koz:AiT90:6:e}, it was shown
                  that in general for linear desynchronized systems there are
                  no algebraic criteria of absolute stability. In this paper,
                  a few misprints occurred in the original version of the
                  article are corrected, and two figures are added.",
}

@ARTICLE{Koz:INFOPROC16:e,
 author        = "Kozyakin, V. S.",
 authauthor    = "Kozyakin, V.",
 title         = "Constructive stability and stabilizability of positive
                  linear discrete-time switching systems",
 journal       = "J. Commun. Technol. Electron.",
 fjournal      = "Journal of Communications Technology and Electronics",
 year          = "2017",
 volume        = "62",
 number        = "6",
 pages         = "686--693",
 eprinttype    = "arXiv",
 eprint        = "1511.05665",
 doi           = "10.1134/S1064226917060110",
 annote        = "We describe a new class of positive linear discrete-time
                  switching systems for which the problems of stability or
                  stabilizability can be resolved constructively. The systems
                  constituting this class can be treated as a natural
                  generalization of systems with the so-called independently
                  switching state vector components. Distinctive feature of
                  such systems is that their components can be arbitrarily
                  ``re-connected'' in parallel or in series without loss of
                  the ``constructive resolvability'' property for the
                  problems of stability or stabilizability of a system. It is
                  shown also that, for such systems, the individual positive
                  trajectories with the greatest or the lowest rate of
                  convergence to the zero can be built constructively.",
}

@ARTICLE{KKP:IFAC94,
 author        = "Kozyakin, V. S. and Kuznetsov, N. A. and Pokrovskii, A. V.",
 authauthor    = "Kozyakin, V. and Kuznetsov, N. and Pokrovskii, A.",
 title         = "Transients in Quasi--Controllable Systems. {O}vershooting,
                  Stability and Instability",
 journal       = "Control Eng. Practice",
 year          = "1994",
 volume        = "2",
 number        = "6",
 pages         = "1080",
 month         = dec,
 eprinttype    = "arXiv",
 eprint        = "0909.4372",
 doi           = "10.1016/0967-0661(94)91809-0",
 annote        = "Families of regimes for control systems are studied
                  possessing the so called quasi-controllability property
                  that is similar to the Kalman controllability property. A
                  new approach is proposed to estimate the degree of
                  transients overshooting in quasi-controllable systems. This
                  approach is conceptually related with the principle of
                  bounded regimes absence in the absolute stability problem.
                  Its essence is in obtaining of constructive a priori bounds
                  for degree of overshooting in terms of the so called
                  quasi-controllability measure. It is shown that relations
                  between stability, asymptotic stability and instability for
                  quasi-controllable systems are similar to those for systems
                  described by linear differential or difference equations in
                  the case when the leading eigenvalue of the corresponding
                  matrix is simple. The results are applicable for analysis
                  of transients, classical absolute stability problem,
                  stability problem for desynchronized systems and so on.",
}

@INPROCEEDINGS{KKP:CESA98,
 author        = "Kozyakin, V. S. and Kuznetsov, N. A. and Pokrovskii, A. V.",
 authauthor    = "Kozyakin, V. and Kuznetsov, N. and Pokrovskii, A.",
 editor        = "Borne, P. and Ksouri, M. and El Kamel, A.",
 title         = "Quasi-Controllability and Estimates of Amplitudes of
                  Transient Regimes in Discrete Systems",
 booktitle     = "Computational Engineering in Systems Applications,
                  {CESA}'98, April 1--4, 1998",
 publisher     = "Ecole Centrale de Lille, France, CD-ROM",
 organization  = "{IMACS}-{IEEE} Multiconference",
 address       = "Hammamet, Tunisia",
 year          = "1998",
 volume        = "1",
 pages         = "266--271",
 eprinttype    = "arXiv",
 eprint        = "0909.4374",
 isbn          = "2-9512309-0-7",
 annote        = "Families of regimes for discrete control systems are studied
                  possessing a special quasi-controllability property that is
                  similar to the Kalman controllability property. A new
                  approach is proposed to estimate the amplitudes of
                  transient regimes in quasi-controllable systems. Its
                  essence is in obtaining of constructive a priori bounds for
                  degree of overshooting in terms of the
                  quasi-controllability measure. The results are applicable
                  for analysis of transients, classical absolute stability
                  problem and, especially, for stability problem for
                  desynchronized systems.",
}

@INPROCEEDINGS{KKV:CESA98,
 author        = "Kozyakin, V. S. and Kuznetsov, N. A. and Vladimirov, A. A.",
 authauthor    = "Kozyakin, V. and Kuznetsov, N. and Vladimirov, A.",
 editor        = "Borne, P. and Ksouri, M. and El Kamel, A.",
 title         = "Matrix methods in {S}korokhod problems",
 booktitle     = "Computational Engineering in Systems Applications,
                  {CESA}'98, April 1--4, 1998",
 publisher     = "Ecole Centrale de Lille, France, CD-ROM",
 organization  = "{IMACS}-{IEEE} Multiconference",
 address       = "Hammamet, Tunisia",
 year          = "1998",
 volume        = "2",
 pages         = "323--328",
 isbn          = "2-9512309-0-7",
 doi           = "10.13140/2.1.4241.1203",
 url           = "https://www.researchgate.net/publication/234109534_Matrix_methods_in_Skorokhod_problems",
 annote        = "The \emph{Skorokhod problem} is a determined mathematical
                  model used for the construction and analysis of constrained
                  processes, both determined and stochastic, such as queing
                  networks, processor sharing in communication networks,
                  stochastic approximation schemes for problems with
                  constraints, etc. The model we deal with consists of a
                  convex polyhedral set $Z$ in a finite-dimensional space
                  $\mathbb{R}^{n}$ and a family of ``reflection vectors''
                  $d_i$ associated with $(n-1)$-dimensional faces $F_i$ of
                  $Z$. According to certain rules of reflection, an output
                  $x(t)\subseteq Z$ is generated for each continuous input
                  $u(t), u(0)\in Z$. The corresponding input-output operator,
                  if it exists, is called the \emph{Skorokhod map}. The
                  properties of the Skorokhod problem such as existence and
                  uniqueness of an output for any admissible input, and
                  different continuity properties of the associated Skorokhod
                  map can be studied in terms of different types of stability
                  of finite families of special $n\times n$-matrices, namely,
                  the projection matrices onto the hyperplanes $L_i$ parallel
                  to faces $F_i$ along the vectors $d_i$. We present both
                  necessary and sufficient conditions of some of the above
                  properties and also establish new relations between such
                  notions as absolute stability, BV-stability, Perron
                  stability, etc., of finite sets of projections and, more
                  generally, of arbitrary $n\times n$-matrices.",
}

@TECHREPORT{KozPok:CADSEM96-005,
 author        = "Kozyakin, V. S. and Pokrovskii, A. V.",
 authauthor    = "Kozyakin, V. and Pokrovskii, A.",
 title         = "Estimates of Amplitudes of Transient Regimes in
                  Quasi-Controllable Discrete Systems",
 institution   = "Deakin University",
 address       = "Geelong, Australia",
 year          = "1996",
 number        = "96--005",
 pagetotal     = "21",
 type          = "CADSEM Report",
 eprinttype    = "arXiv",
 eprint        = "0908.4138",
 annote        = "Families of regimes for discrete control systems are studied
                  possessing a special
                  \emph{quasi--con\-trol\-\l\-abi\-li\-ty property} that is
                  similar to the Kalman con\-trol\-\l\-abi\-li\-ty property.
                  A new approach is proposed to estimate the amplitudes of
                  transient regimes in quasi--controllable systems. Its
                  essence is in obtaining of constructive a priori bounds for
                  degree of overshooting in terms of the
                  quasi--con\-trol\-\l\-abi\-li\-ty measure. The results are
                  applicable for analysis of transients, classical absolute
                  stability problem and, especially, for stability problem
                  for desynchronized systems.",
}

@ARTICLE{KozPok:DAN92:e,
 author        = "Kozyakin, V. S. and Pokrovskij, A. V.",
 authauthor    = "Kozyakin, Victor and Pokrovskii, Alexei",
 title         = "The role of controllability type properties in the study of
                  asynchronous dynamic systems",
 journal       = "Sov. Phys., Dokl.",
 fjournal      = "Soviet Physics. Doklady",
 year          = "1992",
 volume        = "37",
 number        = "5",
 pages         = "213--215",
 issn          = "0038-5689",
 mrclass       = "93B05 (93C05)",
 mrnumber      = "1198568 (93i:93018)",
 zblnumber     = "0790.93018",
 annote        = "Analysis of the stability of so-called synchronous systems
                  has recently attracted considerable attention. In this
                  paper we study the geometric properties of the family of
                  modes for synchronous systems that have special properties
                  of Kalman-controllability type. We show that relations
                  between the properties of stability, asymptotic stability,
                  and instability in the indicated family are analogous to
                  the relations between the corresponding properties for the
                  family of solutions of a linear differential equation or
                  difference equation in the case of simplicity of the
                  leading eigenvalues of the matrices that define these
                  equations. The results stated below are useful for analysis
                  of classical problems on absolute stability. Our approach
                  is similar to the principle of the absence of bounded modes
                  in the problem of absolute stability.",
}

@INPROCEEDINGS{Koz:ICDEA04,
 author        = "Kozyakin, Victor",
 title         = "A short introduction to asynchronous systems",
 booktitle     = "Proceedings of the {S}ixth {I}nternational {C}onference on
                  {D}ifference {E}quations",
 publisher     = "CRC",
 address       = "Boca Raton, FL",
 year          = "2004",
 pages         = "153--165",
 mrclass       = "93A10 (93D05 93D15)",
 mrnumber      = "2092552 (2005e:93003)",
 zblnumber     = "1083.93032",
 zblreviewer   = "Victor Sharapov (Volgograd)",
 annote        = "Looking at the dynamics of the system described by the
                  equation\[ x(n+1)=f(x(n)) \] one may say that coordinates
                  of the vector $x=\{x_{1},x_{2},\ldots,x_{N}\}$ are updated
                  \emph{synchronously}. What happens with the system if
                  coordinates of the vector $x$ are updated
                  \emph{asynchronously}, i.e., if at a given moment $n$ only
                  coordinates with indices $i$ from some set
                  $\omega(n)\subseteq\{1,2,\ldots,N\}$ are changed in
                  accordance with the law\[ x_{i}(n+1)=f_{i}(x(n)) \] while
                  others remain intact? This is the main topic which is
                  discussed below.\par Examples of asynchronous systems are
                  multiprocessor systems, distributed digital networks,
                  discrete-time models of market economy, etc.\par The main
                  attention is be paid to discussion of the problem how
                  asynchronism affects stability of the system. Examples show
                  that all possible combinations of stability/instability for
                  the pair `synchronous/asyn\-chro\-nous system' may occur.
                  And also, simple examples demonstrate that the problem of
                  investigation of stability for asynchronous system is more
                  complicate than for synchronous one, even in linear case.
                  Nevertheless, in some situations asynchronous systems
                  possess more robust properties than synchronous ones.\par
                  Formal explanation of this fact is be presented, and
                  various methods of stability investigation for asynchronous
                  system are discussed.",
}

@ARTICLE{Koz:LAA09,
 author        = "Kozyakin, Victor",
 title         = "On accuracy of approximation of the spectral radius by the
                  {G}elfand formula",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2009",
 volume        = "431",
 number        = "11",
 pages         = "2134--2141",
 eprinttype    = "arXiv",
 eprint        = "0810.2856",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15A18 (15A60)",
 mrnumber      = "2567820 (2011c:15033)",
 mrreviewer    = "Fuad Kittaneh",
 zblnumber     = "1177.15012",
 zblreviewer   = "Grozio Stanilov (Sofia)",
 doi           = "10.1016/j.laa.2009.07.008",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379509003590",
 annote        = "The famous Gelfand formula for the spectral radius of a
                  matrix is of great importance in various mathematical
                  constructions. Unfortunately, the range of applicability of
                  this formula is substantially restricted by a lack of
                  estimates for the rate of convergence of the quantities
                  $\|A^{n}\|^{1/n} \to \rho(A)$. In the paper this deficiency
                  is made up to some extent. By using the Bochi inequalities
                  we establish explicit computable estimates for the rate of
                  convergence of the quantities $\|A^{n}\|^{1/n} \to
                  \rho(A)$. The obtained estimates are then extended for
                  evaluation of the joint spectral radius of matrix sets.",
}

@MISC{Koz:ArXiv10-1,
 author        = "Kozyakin, Victor",
 title         = "Max-{R}elaxation Iteration Procedure for Building of
                  {B}arabanov Norms: Convergence and Examples",
 howpublished  = "ArXiv.org e-Print archive",
 year          = "2010",
 month         = feb,
 eprinttype    = "arXiv",
 eprint        = "1002.3251",
 doi           = "10.48550/arXiv.1002.3251",
 annote        = "The problem of construction of Barabanov norms for analysis
                  of properties of the joint (generalized) spectral radius of
                  matrix sets has been discussed in a number of publications.
                  In earlier author's works the method of Barabanov norms was
                  the key instrument in disproving the Lagarias--Wang
                  Finiteness Conjecture~\cite{LagWang:LAA95}. The related
                  constructions were essentially based on the study of the
                  geometrical properties of the unit balls of some specific
                  Barabanov norms. In this context the situation when one
                  fails to find among current publications any detailed
                  analysis of the geometrical properties of the unit balls of
                  Barabanov norms looks a bit paradoxical. Partially this is
                  explained by the fact that Barabanov norms are defined
                  nonconstructively, by an implicit procedure. So, even in
                  simplest cases it is very difficult to visualize the shape
                  of their unit balls. The present work may be treated as the
                  first step to make up this deficiency. In the paper an
                  iteration procedure is considered that allows to build
                  numerically Barabanov norms for the irreducible matrix sets
                  and simultaneously to compute the joint spectral radius of
                  these sets.",
}

@ARTICLE{Koz:DCDSB10,
 author        = "Kozyakin, Victor",
 title         = "Iterative building of {B}arabanov norms and computation of
                  the joint spectral radius for matrix sets",
 journal       = "Discrete Contin. Dyn. Syst. Ser. B",
 fjournal      = "Discrete and Continuous Dynamical Systems. Series B. A
                  Journal Bridging Mathematics and Sciences",
 year          = "2010",
 volume        = "14",
 number        = "1",
 pages         = "143--158",
 eprinttype    = "arXiv",
 eprint        = "0810.2154",
 issn          = "1531-3492",
 mrclass       = "15A18 (15A60 65F15)",
 mrnumber      = "2644257",
 mrreviewer    = "Panayiotis J. Psarrakos",
 zblnumber     = "1201.65067",
 zblreviewer   = "Georg Hebermehl (Berlin)",
 doi           = "10.3934/dcdsb.2010.14.143",
 url           = "https://www.aimsciences.org/article/doi/10.3934/dcdsb.2010.14.143",
 annote        = "An extremal norm satisfying the following theorem is called
                  a Barabanov
                  norm~\cite{Bar:AIT88-2:e,Bar:AIT88-3:e,Bar:AIT88-5:e}.\par
                  \textbf{Theorem.} \emph{Let the, matrix set $\mathcal{A} =
                  \{A_1,\ldots,A_r\}$ be irreducible. Then the quantity
                  $\rho$ is the joint (generalized) spectral radius of
                  $\mathcal{A}$ if and only if there exist a norm $\|\cdot\|$
                  in $\mathbb{R}^m$ such that $\rho\|x\| = \max\{\|A_1 x\|\},
                  \{\| A_2 x \|,\ldots,\| A_r x \|\}$.}\par The author used
                  this norm in earlier papers (e.g.,
                  \emph{V.~Kozyakin}~\cite{Koz:INFOPROC06:e}), to disprove
                  the Lagarias--Wang finiteness
                  conjecture~\cite{LagWang:LAA95}. The geometrical properties
                  of the unit balls of specific Barabanov norms play a
                  decisive role in the corresponding constructions and in
                  other applications.\par Because the Barabanov norms are
                  only defined by an implicit procedure it is very difficult
                  to visualize the shapes of the related unit balls. Thus, in
                  the present work an iteration procedure is created that
                  allows to build Barabanov norms for irreducible matrix
                  sets. Simultaneously the joint spectral radius of these
                  sets is computed. The convergence of the iteration is
                  formulated in a main theorem and proved with the help of
                  some lemmas. Computational errors of the joint spectral
                  radius can be a posteriori determined. The algorithm is
                  demonstrated by two examples. The corresponding computed
                  unit spheres are shown in figures. Some open problems about
                  the accuracy of the Barabanov norm, about the convergence
                  rate of the joint spectral radius and other are formulated
                  at the end of the paper.",
}

@ARTICLE{Koz:DEDS10,
 author        = "Kozyakin, Victor",
 title         = "On explicit a priori estimates of the joint spectral radius
                  by the generalized {G}elfand formula",
 journal       = "Differ. Equ. Dyn. Syst.",
 fjournal      = "Differential Equations and Dynamical Systems",
 year          = "2010",
 volume        = "18",
 number        = "1-2",
 pages         = "91--103",
 eprinttype    = "arXiv",
 eprint        = "0810.2157",
 issn          = "0971-3514",
 mrclass       = "15A18 (47A10 93D20)",
 mrnumber      = "2670076 (2011f:15032)",
 mrreviewer    = "Janko Bra{\v{c}}i{\v{c}}",
 zblnumber     = "1219.15019",
 zblreviewer   = "Marjeta Kramar Fijav{\v{z}} (Ljubljana)",
 doi           = "10.1007/s12591-010-0010-1",
 url           = "https://link.springer.com/article/10.1007/s12591-010-0010-1",
 annote        = "In various problems of control theory, non-autonomous and
                  multivalued dynamical systems, wavelet theory and other
                  fields of mathematics information about the rate of growth
                  of matrix products with factors taken from some matrix set
                  plays a key role. One of the most prominent quantities
                  characterizing the exponential rate of growth of matrix
                  products is the so-called joint or generalized spectral
                  radius. In the work some explicit a priori estimates for
                  the joint spectral radius with the help of the generalized
                  Gelfand formula are obtained. These estimates are based on
                  the notion of the measure of irreducibility
                  (quasi-controllability) of matrix sets previously
                  introduced and investigated by \emph{V.\,S.~Kozyakin} and
                  \emph{A.\,V.~Pokrovskii}~\cite{KozPok:DAN92:e,KozPok:CADSEM96-005}.",
}

@ARTICLE{Koz:LAA10,
 author        = "Kozyakin, Victor",
 title         = "An explicit {L}ipschitz constant for the joint spectral
                  radius",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2010",
 volume        = "433",
 number        = "1",
 pages         = "12--18",
 eprinttype    = "arXiv",
 eprint        = "0909.3170",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15A60 (15A18)",
 mrnumber      = "2645060 (2011e:15042)",
 mrreviewer    = "Susana M. Furtado",
 zblnumber     = "1198.15006",
 zblreviewer   = "Alan L. Andrew (Bundoora)",
 doi           = "10.1016/j.laa.2010.01.028",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379510000418",
 annote        = "\emph{F.~Wirth}~\cite{Wirth:LAA02} has proved that the joint
                  spectral radius of irreducible compact sets of matrices is
                  locally Lipschitz continuous as a function of the matrix
                  set. In the paper, an explicit formula for the related
                  Lipschitz constant is obtained.",
}

@ARTICLE{Koz:JDEA11,
 author        = "Kozyakin, Victor",
 title         = "A relaxation scheme for computation of the joint spectral
                  radius of matrix sets",
 journal       = "J. Difference Equ. Appl.",
 fjournal      = "Journal of Difference Equations and Applications",
 year          = "2011",
 volume        = "17",
 number        = "2",
 pages         = "185--201",
 eprinttype    = "arXiv",
 eprint        = "0810.4230",
 issn          = "1023-6198",
 mrclass       = "15A18 (15A60 65F15)",
 mrnumber      = "2783343 (2012e:15021)",
 mrreviewer    = "Panayiotis J. Psarrakos",
 zblnumber     = "1214.65015",
 zblreviewer   = "Constantin Popa (Constan{\c{t}}a)",
 doi           = "10.1080/10236198.2010.549008",
 url           = "https://www.tandfonline.com/doi/abs/10.1080/10236198.2010.549008",
 annote        = "The problem of computation of the joint (generalized)
                  spectral radius of matrix sets has been discussed in a
                  number of publications. In this paper, an iteration
                  procedure is proposed that allows to build numerically
                  Barabanov norms for the irreducible matrix sets and
                  simultaneously to compute the joint spectral radius of
                  these sets.",
}

@ARTICLE{Koz:LAA14,
 author        = "Kozyakin, Victor",
 title         = "The {B}erger-{W}ang formula for the {M}arkovian joint
                  spectral radius",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2014",
 volume        = "448",
 pages         = "315--328",
 month         = may,
 eprinttype    = "arXiv",
 eprint        = "1401.2711",
 issn          = "0024-3795",
 mrclass       = "15A60 (60J20)",
 mrnumber      = "3182989",
 mrreviewer    = "Jaros{\l}aw Adam Miszczak",
 zblnumber     = "06278067",
 doi           = "10.1016/j.laa.2014.01.022",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379514000408",
 annote        = "The Berger--Wang formula establishes equality between the
                  joint and generalized spectral radii of a set of matrices.
                  For matrix products whose multipliers are applied not
                  arbitrarily but in accordance to some Markovian law, there
                  are also known analogs of the joint and generalized
                  spectral radii. However, the known proofs of the
                  Berger--Wang formula hardly can be directly applied in the
                  case of Markovian products of matrices since they
                  essentially rely on the arbitrariness of appearance of
                  different matrices in the related matrix products.
                  Nevertheless, as has been shown by
                  \emph{X.~Dai}~\cite{Dai:JFI14} the Berger--Wang formula is
                  valid for the case of Markovian analogs of the joint and
                  the generalized spectral radii too, although the proof in
                  this case heavily exploits the more involved techniques of
                  multiplicative ergodic theory. In the paper we propose a
                  matrix theory construction allowing deduce the Markovian
                  analog of the Berger--Wang formula from the classical
                  Berger--Wang formula.",
}

@ARTICLE{Koz:LAA14-2,
 author        = "Kozyakin, Victor",
 title         = "Matrix products with constraints on the sliding block
                  relative frequencies of different factors",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2014",
 volume        = "457",
 pages         = "244--260",
 month         = sep,
 eprinttype    = "arXiv",
 eprint        = "1403.5050",
 issn          = "0024-3795",
 mrclass       = "15A60 (37B10 60J10)",
 mrnumber      = "3230443",
 zblnumber     = "1291.15027",
 doi           = "10.1016/j.laa.2014.05.016",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379514002961",
 annote        = "One of fundamental results of the theory of
                  joint/generalized spectral radius, the Berger--Wang
                  theorem, establishes equality between the joint and
                  generalized spectral radii of a set of matrices.
                  Generalization of this theorem on products of matrices
                  whose factors are applied not arbitrarily but are subjected
                  to some constraints is connected with essential
                  difficulties since known proofs of the Berger--Wang theorem
                  rely on the arbitrariness of appearance of different
                  matrices in the related matrix products. Recently,
                  \emph{X.~Dai}~\cite{Dai:JFI14} proved an analog of the
                  Berger--Wang theorem for the case when factors in matrix
                  products are formed by some Markov law.\par We introduce
                  the concepts of the joint and generalized spectral radii
                  for products of matrices subjected to constraints on the
                  sliding block relative frequencies of occurrences of
                  different matrices, and prove an analog of the Berger--Wang
                  theorem for this case.",
}

@ARTICLE{Koz:LAA16,
 author        = "Kozyakin, Victor",
 title         = "Hourglass alternative and the finiteness conjecture for the
                  spectral characteristics of sets of non-negative matrices",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2016",
 volume        = "489",
 pages         = "167--185",
 eprinttype    = "arXiv",
 eprint        = "1507.00492",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15B48 (15A18 15A60)",
 mrnumber      = "3421844",
 mrreviewer    = "Ravindra B. Bapat",
 zblnumber     = "1326.15053",
 doi           = "10.1016/j.laa.2015.10.017",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379515006126",
 annote        = "Recently Blondel, Nesterov and Protasov
                  proved~\cite{BN:SIAMJMAA09,NesPro:SIAMJMAA13} that the
                  finiteness conjecture holds for the generalized and the
                  lower spectral radii of the sets of non-negative matrices
                  with independent row/column uncertainty. We show that this
                  result can be obtained as a simple consequence of the
                  so-called hourglass alternative earlier used by the author
                  and his companions to analyze the minimax relations between
                  the spectral radii of matrix products. Axiomatization of
                  the statements that constitute the hourglass alternative
                  makes it possible to define a new class of sets of positive
                  matrices having the finiteness property, which includes the
                  sets of non-negative matrices with independent row
                  uncertainty. This class of matrices, supplemented by the
                  zero and identity matrices, forms a semiring with the
                  Minkowski operations of addition and multiplication of
                  matrix sets, which gives means to construct new sets of
                  non-negative matrices possessing the finiteness property
                  for the generalized and the lower spectral radii.",
}

@CONFERENCE{Koz:WSDV16,
 author        = "Kozyakin, Victor",
 title         = "Hourglass Alternative and constructivity of spectral
                  characteristics of matrix products",
 booktitle     = "Workshop on switching dynamics {\&} verification, January
                  28-29",
 organization  = "Institut pour le Contr{\^o}le et la D{\'e}cision de l'Idex
                  Paris-Saclay, Paris, France",
 year          = "2016",
 doi           = "10.13140/RG.2.1.4880.1042",
 url           = "https://www.researchgate.net/publication/294368448_Hourglass_Alternative_and_constructivity_of_spectral_characteristics_of_matrix_products?channel=doi&linkId=56c0324708ae44da37fa8654&showFulltext=true",
 language      = "english",
 annote        = "Recently Blondel, Nesterov and Protasov proved that the
                  finiteness conjecture holds for the generalized and the
                  lower spectral radii of the sets of non-negative matrices
                  with independent row/column uncertainty. We show that this
                  result can be obtained as a simple consequence of the
                  so-called hourglass alternative which can be used also to
                  establish the minimax relations between the spectral radii
                  of matrix products. Axiomatization of the statements that
                  constitute the hourglass alternative makes it possible to
                  define a new class of sets of positive matrices having the
                  finiteness property, which includes the sets of
                  non-negative matrices with independent row uncertainty.
                  This class of matrices, supplemented by the zero and
                  identity matrices, forms a semiring with the Minkowski
                  operations of addition and multiplication of matrix sets,
                  which gives means to construct new sets of non-negative
                  matrices possessing the finiteness property for the
                  generalized and the lower spectral radii.\par The Hourglass
                  alternative helps us to describe a new class of positive
                  linear discrete-time switching systems for which the
                  problems of stability or stabilizability can be resolved
                  constructively. This class generalizes the class of systems
                  with independently switching state vector components. The
                  distinctive feature of this class is that, for any system
                  from this class, its components or blocks can be
                  arbitrarily connected in parallel or in series without loss
                  of the `constructive resolvability' property. It is shown
                  also that, for such systems, it is possible to build
                  constructively the individual positive trajectories with
                  the greatest or the lowest rate of convergence to the
                  zero.",
}

@ARTICLE{Koz:LMA17,
 author        = "Kozyakin, Victor",
 title         = "Minimax theorem for the spectral radius of the product of
                  non-negative matrices",
 journal       = "Linear Multilinear Algebra",
 fjournal      = "Linear and Multilinear Algebra",
 year          = "2017",
 volume        = "65",
 number        = "11",
 pages         = "2356--2365",
 eprinttype    = "arXiv",
 eprint        = "1603.05375",
 issn          = "0308-1087",
 mrclass       = "15A45 (15A60 15B48 49J35)",
 mrnumber      = "3740702",
 mrreviewer    = "Jaspal Singh Aujla",
 zblnumber     = "1387.15017",
 doi           = "10.1080/03081087.2016.1273877",
 url           = "https://www.tandfonline.com/doi/abs/10.1080/03081087.2016.1273877",
 language      = "english",
 annote        = "We prove the minimax equality for the spectral radius
                  $\rho(AB)$ of the product of matrices $A\in\mathcal{A}$ and
                  $B\in\mathcal{B}$, where $\mathcal{A}$ and $\mathcal{B}$
                  are compact sets of non-negative matrices of dimensions
                  $N\times M$ and $M\times N$, respectively, satisfying the
                  so-called hourglass alternative.",
}

@ARTICLE{Koz:DDNS18,
 author        = "Kozyakin, Victor",
 title         = "On convergence of infinite matrix products with alternating
                  factors from two sets of matrices",
 journal       = "Discrete Dyn. Nat. Soc.",
 fjournal      = "Discrete Dynamics in Nature and Society",
 year          = "2018",
 volume        = "2018",
 number        = "9216760",
 pages         = "5",
 eprinttype    = "arXiv",
 eprint        = "1712.06356",
 issn          = "1026-0226",
 mrclass       = "93D20 (15A60 40A20)",
 mrnumber      = "3798756",
 mrreviewer    = "Roksana Krystyna S{\l o}wik",
 zblnumber     = "1417.93254",
 doi           = "10.1155/2018/9216760",
 url           = "https://www.hindawi.com/journals/ddns/2018/9216760",
 annote        = "We consider the problem of convergence to zero of matrix
                  products $A_{n}B_{n}\cdots A_{1}B_{1}$ with factors from
                  two sets of matrices, $A_{i}\in\mathcal{A}$ and
                  $B_{i}\in\mathcal{B}$, due to a suitable choice of matrices
                  $\{B_{i}\}$. It is assumed that for any sequence of
                  matrices $\{A_{i}\}$ there is a sequence of matrices
                  $\{B_{i}\}$ such that the corresponding matrix product
                  $A_{n}B_{n}\cdots A_{1}B_{1}$ converges to zero. We show
                  that in this case the convergence of the matrix products
                  under consideration is uniformly exponential, that is,
                  $\|A_{n}B_{n}\cdots A_{1}B_{1}\|\le C\lambda^{n}$, where
                  the constants $C>0$ and $\lambda\in(0,1)$ do not depend on
                  the sequence $\{A_{i}\}$ and the corresponding sequence
                  $\{B_{i}\}$.\par Other problems of this kind are discussed
                  and open questions are formulated.",
}

@ARTICLE{Koz:DCDSB19,
 author        = "Kozyakin, Victor",
 title         = "Minimax joint spectral radius and stabilizability of
                  discrete-time linear switching control systems",
 journal       = "Discrete Contin. Dyn. Syst. Ser. B",
 fjournal      = "Discrete and Continuous Dynamical Systems. Series B. A
                  Journal Bridging Mathematics and Sciences",
 year          = "2019",
 volume        = "24",
 number        = "8",
 pages         = "3537--3556",
 eprinttype    = "arXiv",
 eprint        = "1712.06805",
 issn          = "1531-3492",
 mrclass       = "93D15 (15A18 93B25 93C30 93C55 94C11)",
 mrnumber      = "3986244",
 zblnumber     = "1427.93180",
 doi           = "10.3934/dcdsb.2018277",
 url           = "https://aimsciences.org/article/doi/10.3934/dcdsb.2018277",
 annote        = "To estimate the growth rate of matrix products $A_{n}\cdots
                  A_{1}$ with factors from some set of matrices
                  $\mathscr{A}$, such numeric quantities as the joint
                  spectral radius $\rho(\mathscr{A})$ and the lower spectral
                  radius $\check{\rho}(\mathscr{A})$ are traditionally used.
                  The first of these quantities characterizes the maximum
                  growth rate of the norms of the corresponding products,
                  while the second one characterizes the minimal growth rate.
                  In the theory of discrete-time linear switching systems,
                  the inequality $\rho(\mathscr{A})<1$ serves as a criterion
                  for the stability of a system, and the inequality
                  $\check{\rho}(\mathscr{A})<1 $ as a criterion for
                  stabilizability.\par For matrix products $A_{n}B_{n}\cdots
                  A_{1}B_{1}$ with factors $A_{i}\in\mathscr{A}$ and
                  $B_{i}\in\mathscr{B}$, where $\mathscr{A}$ and
                  $\mathscr{B}$ are some sets of matrices, we introduce the
                  quantities $\mu(\mathscr{A},\mathscr{B})$ and
                  $\eta(\mathscr{A},\mathscr{B})$, called the lower and upper
                  minimax joint spectral radius of the pair
                  $\{\mathscr{A},\mathscr{B}\}$, respectively, which
                  characterize the maximum growth rate of the matrix products
                  $A_{n}B_{n}\cdots A_{1}B_{1}$ over all sets of matrices
                  $A_{i}\in\mathscr{A}$ and the minimal growth rate over all
                  sets of matrices $B_{i}\in\mathscr{B}$. In this sense, the
                  minimax joint spectral radii can be considered as
                  generalizations of both the joint and lower spectral radii.
                  As an application of the minimax joint spectral radii, it
                  is shown how these quantities can be used to analyze the
                  stabilizability of discrete-time linear switching control
                  systems in the presence of uncontrolled external
                  disturbances of the plant.",
}

@MISC{Koz:ArXiv20,
 author        = "Kozyakin, Victor",
 title         = "On the boundedness of infinite matrix products with
                  alternating factors from two sets of matrices",
 howpublished  = "ArXiv.org e-Print archive",
 year          = "2020",
 month         = oct,
 eprinttype    = "arXiv",
 eprint        = "2010.03890",
 doi           = "10.48550/arXiv.2010.03890",
 language      = "english",
 annote        = "We consider the question of boundedness of matrix products
                  $A_{n}B_{n}\cdots A_{1}B_{1}$ with factors from two sets of
                  matrices, $A_{i}\in\mathcal{A}$ and $B_{i}\in\mathcal{B}$,
                  due to an appropriate choice of matrices $\{B_{i}\}$. It is
                  assumed that for every sequence of matrices $\{A_{i}\}$
                  there is a sequence of matrices $\{B_{i}\}$ for which the
                  sequence of matrix products $\{A_{n}B_{n}\cdots
                  A_{1}B_{1}\}_{n=1}^{\infty}$ is norm bounded. Some
                  situations are described where in this case the norms of
                  the matrix products $A_{n}B_{n}\cdots A_{1}B_{1}$ are
                  uniformly bounded, that is, $\|A_{n}B_{n}\cdots
                  A_{1}B_{1}\|\le C$ for all natural numbers $n$, where $C >
                  0$ is a constant independent of the sequence $\{A_{i}\}$
                  and the corresponding sequence $\{B_{i}\}$. For the general
                  case, the question of the validity of the corresponding
                  statement remains open.",
}

@ARTICLE{Koz:Automatica22,
 author        = "Kozyakin, Victor",
 title         = "Non-{S}turmian sequences of matrices providing the maximum
                  growth rate of matrix products",
 journal       = "Automatica J. IFAC",
 fjournal      = "Automatica. A Journal of IFAC, the International Federation
                  of Automatic Control",
 volume        = "145",
 year          = "2022",
 month         = nov,
 pages         = "Paper No. 110574, 10",
 issn          = "0005-1098",
 mrclass       = "15A60 (37C35 93C05 93C30)",
 mrnumber      = "4473668",
 zblnumber     = "07595547",
 eprinttype    = "arXiv",
 eprint        = "2112.00391",
 doi           = "10.1016/j.automatica.2022.110574",
 url           = "https://www.sciencedirect.com/science/article/pii/S0005109822004356",
 language      = "english",
 annote        = "In the theory of linear switching systems with discrete
                  time, as in other areas of mathematics, the problem of
                  studying the growth rate of the norms of all possible
                  matrix products $A_{\sigma_{n}}\cdots A_{\sigma_{0}}$ with
                  factors from a set of matrices $\mathcal{A}$ arises. So
                  far, only for a relatively small number of classes of
                  matrices $\mathcal{A}$ has it been possible to accurately
                  describe the sequences of matrices that guarantee the
                  maximum rate of increase of the corresponding norms.
                  Moreover, in almost all cases studied theoretically, the
                  index sequences $\{\sigma_{n}\}$ of matrices maximizing the
                  norms of the corresponding matrix products have been shown
                  to be periodic or so-called Sturmian, which entails a whole
                  set of ``good'' properties of the sequences
                  $\{A_{\sigma_{n}}\}$, in particular the existence of a
                  limiting frequency of occurrence of each matrix factor
                  $A_{i}\in\mathscr{A}$ in them. In the paper it is shown
                  that this is not always the case: a class of matrices is
                  defined consisting of two $2\times 2$ matrices, similar to
                  rotations in the plane, in which the sequence
                  $\{A_{\sigma_{n}}\}$ maximizing the growth rate of the
                  norms $\|A_{\sigma_{n}}\cdots A_{\sigma_{0}}\|$ is not
                  Sturmian. All considerations are based on numerical
                  modeling and cannot be considered mathematically rigorous
                  in this part; rather, they should be interpreted as a set
                  of questions for further comprehensive theoretical
                  analysis.",
}

@MISC{KozCalPok:ArXiv10,
 author        = "Kozyakin, Victor and O'Callaghan, Brian and Pokrovskii,
                  Alexei",
 title         = "Sequences of Arbitrages",
 howpublished  = "ArXiv.org e-Print archive",
 year          = "2010",
 month         = apr,
 eprinttype    = "arXiv",
 eprint        = "1004.0561",
 doi           = "10.48550/arXiv.1004.0561",
 annote        = "The goal of this article is to understand some interesting
                  features of sequences of arbitrage operations, which look
                  relevant to various processes in Economics and Finances. In
                  the second part of the paper, analysis of sequences of
                  arbitrages is reformulated in the linear algebra terms.
                  This admits an elegant geometric interpretation of the
                  problems under consideration linked to the asynchronous
                  systems theory. We feel that this interpretation will be
                  useful in understanding more complicated, and more
                  realistic, mathematical models in economics.",
}

@INPROCEEDINGS{LJP:HSCC16,
 author        = "Legat, Beno{\^\i}t and Jungers, Rapha{\"e}l M. and Parrilo,
                  Pablo A.",
 authauthor    = "Legat, Beno{\^\i}t and Jungers, Rapha{\"e}l and Parrilo,
                  Pablo",
 title         = "Generating Unstable Trajectories for Switched Systems via
                  Dual Sum-Of-Squares Techniques",
 booktitle     = "Proceedings of the 19th International Conference on Hybrid
                  Systems: Computation and Control",
 series        = "HSCC '16",
 publisher     = "ACM",
 address       = "New York, NY, USA",
 location      = "Vienna, Austria",
 year          = "2016",
 pages         = "51--60",
 pagetotal     = "10",
 isbn          = "978-1-4503-3955-1",
 mrclass       = "15A99 (39A30 68Q45 90C90)",
 mrnumber      = "3578686",
 zblnumber     = "1364.93315",
 acmid         = "2883821",
 doi           = "10.1145/2883817.2883821",
 url           = "https://dl.acm.org/doi/10.1145/2883817.2883821",
 annote        = "The joint spectral radius (JSR) of a set of matrices
                  characterizes the maximal asymptotic growth rate of an
                  infinite product of matrices of the set. This quantity
                  appears in a number of applications including the stability
                  of switched and hybrid systems. Many algorithms exist for
                  estimating the JSR but not much is known about how to
                  generate an infinite sequence of matrices with an optimal
                  asymptotic growth rate. To the best of our knowledge, the
                  currently known algorithms select a small sequence with
                  large spectral radius using brute force (or
                  branch-and-bound variants) and repeats this sequence
                  infinitely.\par In this paper we introduce a new approach
                  to this question, using the dual solution of a sum of
                  squares optimization program for JSR approximation. Our
                  algorithm produces an infinite sequence of matrices with an
                  asymptotic growth rate arbitrarily close to the JSR. The
                  algorithm naturally extends to the case where the allowable
                  switching sequences are determined by a graph or finite
                  automaton. Unlike the brute force approach, we provide a
                  guarantee on the closeness of the asymptotic growth rate to
                  the JSR. This, in turn, provides new bounds on the quality
                  of the JSR approximation. We provide numerical examples
                  illustrating the good performance of the algorithm.",
}

@ARTICLE{LPG:SIAM20,
 author        = "Legat, Beno{\^\i}t and Parrilo, Pablo and Jungers,
                  Rapha{\"e}l",
 authauthor    = "Beno{\^{i}}t Legat and Parrilo, Pablo and Rapha{\"e}l
                  Jungers",
 title         = "Certifying unstability of switched systems using sum of
                  squares programming",
 journal       = "SIAM J. Control Optim.",
 fjournal      = "SIAM Journal on Control and Optimization",
 year          = "2020",
 volume        = "58",
 number        = "4",
 pages         = "2616--2638",
 issn          = "0363-0129",
 mrclass       = "93D20 (15A60 90C90)",
 mrnumber      = "4142040",
 zblnumber     = "07268459",
 doi           = "10.1137/18M1173460",
 url           = "https://epubs.siam.org/doi/abs/10.1137/18M1173460",
 annote        = "The joint spectral radius (JSR) of a set of matrices
                  characterizes the maximal asymptotic growth rate of an
                  infinite product of matrices of the set. This quantity
                  appears in a number of applications including the stability
                  of switched and hybrid systems. A popular method used for
                  the stability analysis of these systems searches for a
                  Lyapunov function with convex optimization tools.\par We
                  investigate dual formulations for this approach and
                  leverage these dual programs for developing new analysis
                  tools for the JSR. We show that the dual of this convex
                  problem searches for the occupations measures of
                  trajectories with high asymptotic growth rate. We both show
                  how to generate a sequence of guaranteed high asymptotic
                  growth rate and how to detect cases where we can provide
                  lower bounds to the JSR. We deduce from it a new guarantee
                  for the upper bound provided by the sum of squares lyapunov
                  program. We end this paper with a method to reduce the
                  computation of the JSR of low rank matrices to the
                  computation of the constrained JSR of matrices of small
                  dimension.\par All results of this paper are presented for
                  the general case of constrained switched systems, that is,
                  systems for which the switching signal is constrained by an
                  automaton.",
}

@ARTICLE{Leiz:LAA92,
 author        = "Leizarowitz, Arie",
 title         = "On infinite products of stochastic matrices",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "1992",
 volume        = "168",
 pages         = "189--219",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15A51 (60G07 93E20)",
 mrnumber      = "1154468 (93b:15023)",
 mrreviewer    = "Winston T. Lin",
 zblnumber     = "0748.15023",
 zblreviewer   = "R. Sinkhorn (Houston)",
 doi           = "10.1016/0024-3795(92)90294-K",
 url           = "https://www.sciencedirect.com/science/article/pii/002437959290294K",
 annote        = "For a given sequence $\{Q_0,Q_1,\ldots\}$ of $n\times n$
                  stochastic matrices let $\{T_0,T_1,\ldots\}$ be a sequence
                  of products taken in a certain order of multiplication. One
                  says that weak ergodicity obtains for this order of
                  multiplication if $(T_N)_{ik}-(T_N)_{jk}\to 0$ as
                  $N\to\infty$ for every $i,j$ and $k$. One says that strong
                  ergodicity obtains for this order of multiplication if weak
                  ergodicity obtains and $(T_N)_{ij}$ converges to a limit as
                  $N\to\infty$ for every $i$ and $j$.\par Conditions are
                  established for weak ergodicity of products taken in an
                  arbitrary order, and strong ergodicity of the backward
                  products $M_N=Q_N Q_{N-1}\cdots Q_1 Q_0$. Strong ergodicity
                  is studied for products taken in an arbitrary order.",
}

@ARTICLE{Li:IEEETAC11,
 author        = "Li, Keyong",
 title         = "Theories and Ultra Efficient Computation of Joint Spectral
                  Radius for Estimating First Passage Time Distribution of
                  {M}arkov Set-Chain",
 journal       = "{IEEE} Transactions on Automatic Control",
 year          = "2011",
 volume        = "56",
 number        = "12",
 pages         = "2951--2956",
 issn          = "0018-9286",
 mrclass       = "60J10 (15B51)",
 mrnumber      = "2906684",
 zblnumber     = "1368.60076",
 doi           = "10.1109/TAC.2011.2161791",
 url           = "https://ieeexplore.ieee.org/document/5953481",
 annote        = "This technical note is concerned with the tail distribution
                  of the first passage time of Markov set chains (MSC). An
                  original two-part idea---a more progressive relation and a
                  sortedness test---is conceived to characterize such chains.
                  The theoretical construction based on this idea further
                  results in an algorithm that can compute the tightest
                  exponent bound of the tail distribution for
                  high-dimensional problem instances with surprising ease. To
                  understand the computational implication of this algorithm,
                  note that the problem is equivalent to computing the joint
                  spectral radius (JSR) of a special independent column
                  polytope (one that defines Markov set chains) of
                  nonnegative matrices. In this context, the reported
                  algorithm can compute the exact JSR value for cases of
                  $100\times{}100$ matrices in less than a second in Matlab.
                  Problems of this size is far beyond the scope of known JSR
                  techniques. It is worth noting that the fields of MSC and
                  JSR have not had significant overlap as one may expect,
                  despite their conceptual akiness. Meanwhile, the present
                  technical note is a contribution that belongs to both
                  fields.",
}

@ARTICLE{LS:LAA02,
 author        = "Li, Yuan-Chuan and Shih, Mau-Hsiang",
 title         = "Contractibility of compact contractions in {H}ilbert space",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2002",
 volume        = "341",
 pages         = "369--378",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "47A12 (15A18 47A30 47B99)",
 mrnumber      = "1873634 (2003a:47010)",
 zblnumber     = "0999.15010",
 zblreviewer   = "Jitka Drko{\v{s}}ova (Praha)",
 doi           = "10.1016/S0024-3795(01)00489-X",
 url           = "https://www.sciencedirect.com/science/article/pii/S002437950100489X",
 note          = "Special issue dedicated to Professor T.~Ando",
 annote        = "For a \emph{finite} set $\Sigma$ of compact contractions in
                  a complex Hilbert space $(H,\|\cdot\|)$, it is shown that
                  $r(A)<1$ for all $A$ in the multiplicative semigroup
                  generated by $\Sigma$ if and only if there exists a
                  positive integer $N$ such that $\|A\|<1$ for all $A$ in the
                  multiplicative semigroup generated by $\Sigma$ with length
                  greater than $N$. Here $r(A)$ denotes the spectral radius
                  of $A$. As an application, an answer is given to an
                  infinite-dimensional case of the finiteness conjecture for
                  the generalized spectral radius attributed to
                  \emph{J.\,C.~Lagarias} and
                  \emph{Y.~Wang}~\cite{LagWang:LAA95}.",
}

@ARTICLE{LiShih:LAA08,
 author        = "Li, Yuan-Chuan and Shih, Mau-Hsiang",
 title         = "The normed finiteness property of compact contraction
                  operators",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2008",
 volume        = "428",
 number        = "10",
 pages         = "2319--2323",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "47A30 (15A18 47A10 47A13)",
 mrnumber      = "2405248 (2009k:47031)",
 zblnumber     = "1162.47011",
 zblreviewer   = "Edward L. Pekarev (Odessa)",
 doi           = "10.1016/j.laa.2007.08.030",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379507003710",
 annote        = "We study the joint spectral radius given by a finite set of
                  compact operators on a Hilbert space. It is shown that the
                  normed finiteness property holds in this case, that is, if
                  all the compact operators are contractions and the joint
                  spectral radius is equal to 1 then there exists a finite
                  product that has a spectral radius equal to 1. We prove an
                  additional statement in that the requirement that the joint
                  spectral radius be equal to 1 can be relaxed to the asking
                  that the maximum norm of finite products of a length norm
                  is equal to 1. The length of this product is related to the
                  dimension of the subspace on which the set of operators is
                  norm preserving.\par Specifically, let $\Sigma$ be a
                  nonempty finite set of (linear) compact contraction
                  operators on a Hilbert space $\mathcal{H}$, let
                  $\Sigma^{m}$ $(m=1,2,\ldots)$ be the set of all products of
                  operators in $\Sigma$ of length $m$, and let
                  $\widetilde{\rho}(\Sigma)$ denote the joint spectral radius
                  of $\Sigma$, that is, $\widetilde{\rho}(\Sigma) =
                  \limsup_{m\rightarrow \infty}[\sup{(\|A\| :
                  A\in\Sigma^{m})}]^{1/m}$. The authors prove for such
                  $\Sigma$ that $\|Ax\|<\|x\|$ with some $A\in\Sigma^{m}$ and
                  $(0\neq)x\in\mathcal{H}$, that if
                  $\widetilde{\rho}(\Sigma)=1$ or there exists a finite
                  product of the unit norm, then there exists a finite
                  product with spectral radius equal to $1$.",
}

@ARTICLE{LimaRah:JPA94,
 author        = "Lima, R. and Rahibe, M.",
 title         = "Exact {L}yapunov exponent for infinite products of random
                  matrices",
 journal       = "J. Phys. A",
 fjournal      = "Journal of Physics. A. Mathematical and General",
 year          = "1994",
 volume        = "27",
 number        = "10",
 pages         = "3427--3437",
 eprinttype    = "arXiv",
 eprint        = "chao-dyn/9407013",
 coden         = "JPHAC5",
 issn          = "0305-4470",
 mrclass       = "82B05 (15A52 60B15 60J15 82B41 82B44)",
 mrnumber      = "1282183 (95d:82004)",
 mrreviewer    = "Estelle L. Basor",
 zblnumber     = "0830.15018",
 zblreviewer   = "Yang Yingchen (Beijing)",
 doi           = "10.1088/0305-4470/27/10/019",
 url           = "https://iopscience.iop.org/article/10.1088/0305-4470/27/10/019",
 annote        = "Despite significant work since the original paper by
                  H.~Furstenberg (1963), explicit formulae for Lyapunov
                  exponents of infinite products of random matrices are
                  available only in very few cases. In this work, we get a
                  rigorous explicit formula for the Lyapunov exponent for
                  some binary infinite products of random $2\times 2$ real
                  matrices. All these products are constructed using only two
                  types of matrices, $A$ and $B$, which are chosen according
                  to a stochastic process. The matrix $A$ is singular,
                  namely, its determinant is zero. This formula is derived by
                  using a particular decomposition for the matrix $B$, which
                  allows us to write the Lyapunov exponent as a sum of
                  convergent series. The key point is the computation of all
                  integer powers of $B$, which is achieved by a suitable
                  change of frame. The computation then follows by looking at
                  each of the special types of $B$ (hyperbolic, parabolic and
                  elliptic). Finally, we show with an example, that the
                  Lyapunov exponent is a discontinuous function of the given
                  parameter.",
}

@INCOLLECTION{LiuXiao:LNCS12,
 author        = "Liu, Jun and Xiao, Mingqing",
 title         = "Computation of Joint Spectral Radius for Network Model
                  Associated with Rank-One Matrix Set",
 booktitle     = "Neural Information Processing. Proceedings of the 19th
                  International Conference, ICONIP 2012, Doha, Qatar,
                  November 12-15, 2012, Part III",
 series        = "Lecture Notes in Computer Science",
 publisher     = "Springer Berlin Heidelberg",
 year          = "2012",
 volume        = "7665",
 pages         = "356--363",
 isbn          = "978-3-642-34486-2",
 doi           = "10.1007/978-3-642-34487-9_44",
 url           = "https://link.springer.com/chapter/10.1007/978-3-642-34487-9_44",
 annote        = "In the paper, we prove that any finite set of rank-one
                  matrices has the finiteness property by making use of
                  (invariant) extremal norm. An explicit formula for the
                  computation of joint/generalized spectral radius of such
                  type of matrix sets is derived. Several numerical examples
                  from current literature are provided to illustrate our
                  theoretical conclusion.",
}

@ARTICLE{JM:LAA13,
 author        = "Liu, Jun and Xiao, Mingqing",
 title         = "Rank-one characterization of joint spectral radius of finite
                  matrix family",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2013",
 volume        = "438",
 number        = "8",
 pages         = "3258--3277",
 eprinttype    = "arXiv",
 eprint        = "1109.1356",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15A60",
 mrnumber      = "3023275",
 zblnumber     = "1267.15009",
 zblreviewer   = "John D. Dixon (Ottawa)",
 doi           = "10.1016/j.laa.2012.12.032",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379513000190",
 annote        = "Let $\mathcal{F}$ be a finite set of $n\times n$ complex
                  matrices and for each $k\geq1$ let $\mathcal{F}_{k}$ denote
                  the set of all products $A_{1}\cdots A_{k}$ of length $k$
                  with each $A_{i}\in\mathcal{F}$. The joint spectral radius
                  of $\mathcal{F}$ is defined as $\rho(\mathcal{F}):=
                  \inf_{\left\|~\right\|}\max_{A\in\mathcal{F}}\left\|A\right\|$
                  where the infimum is taken over all sub-multiplicative
                  matrix norms. The authors are interested in methods of
                  computing $\rho(\mathcal{F})$.\par It can be shown that
                  \[\max_{A\in\mathcal{F}_{k}}\rho(A)^{1/k}\leq
                  \rho(\mathcal{F})\leq\max_{A\in\mathcal{F}_{k}}
                  \left\|A\right\|^{1/k}\text{ for all }k\geq1\] where
                  $\rho(A)$ denotes the spectral radius of $A$ and
                  $\left\|~\right\|$ is any sub-multiplicative norm.
                  Furthermore, the right hand side converges to
                  $\rho(\mathcal{F})$ as $k\rightarrow\infty$ (see
                  \emph{G.-C.~Rota} and \emph{G.~Strang}~\cite{RotaStr:IM60})
                  and the $\limsup$ of the left hand side is also equal to
                  $\rho(\mathcal{F})$. The set $\mathcal{F}$ is said to have
                  the finiteness property if $\rho(\mathcal{F})$ is equal to
                  $\rho(A)^{1/k}$ for some $k$ and some
                  $A\in\mathcal{F}_{k}$. Although not every finite set
                  $\mathcal{F}$ has the finiteness property, there are some
                  important classes of sets of matrices which do have this
                  property, and when this holds there may be more efficient
                  ways to compute the value of $\rho(\mathcal{F})$.\par The
                  first part of the present paper proves that if
                  $\mathcal{F}$ contains at most one matrix of rank $>1$,
                  then $\mathcal{F}$ has the finiteness property. Moreover,
                  if all matrices in $\mathcal{F}$ have rank $1$ and $k$ is
                  the least value for which there there exists
                  $A\in\mathcal{F}_{k}$ such that
                  $\rho(\mathcal{F})=\rho(A)^{1/k}$, then $A$ is a product of
                  $k$ distinct matrices from $\mathcal{F}$. In this special
                  case, for small sets $\mathcal{F}$, this gives an efficient
                  way to compute $\rho(\mathcal{F})$ (see also
                  \emph{A.\,A.~Ahmadi} and
                  \emph{P.\,A.~Parrilo}~\cite{AhmPar:CDC12}). To extend
                  beyond this limited case, the authors consider the set
                  $P(\mathcal{F}_{k})$ of rank one approximations to the
                  elements in $\mathcal{F}_{k}$ (which may be obtained using
                  singular value decompositions) and prove that
                  $\rho(\mathcal{F})=
                  \lim\sup_{k\rightarrow\infty}\rho(P(\mathcal{F}_{k}))^{1/k}$.
                  In the special case where each matrix in $\mathcal{F}$ has
                  nonnegative real entries and some product of matrices in
                  $\mathcal{F}$ has all entries $>0$, there is a simpler
                  formula: $\rho(\mathcal{F})$ is equal to the limit of
                  $\max_{A\in\mathcal{F}_{k}}~(\text{tr}~A)^{1/k}$ as
                  $k\rightarrow\infty$. Numerical examples are given to show
                  how these formulae may be used in practice.",
}

@ARTICLE{LMMR:JOT98,
 author        = "Livshits, L. and MacDonald, G. and Mathes, B. and Radjavi,
                  H.",
 title         = "Reducible semigroups of idempotent operators",
 journal       = "J. Operator Theory",
 fjournal      = "Journal of Operator Theory",
 year          = "1998",
 volume        = "40",
 number        = "1",
 pages         = "35--69",
 issn          = "0379-4024",
 mrclass       = "47A15 (20M99 47D03)",
 mrnumber      = "1642522 (99m:47004)",
 mrreviewer    = "Cecelia Laurie",
 zblnumber     = "0995.47002",
 url           = "https://www.theta.ro/jot/archive/1998-040-001/1998-040-001-002.html",
 annote        = "We study the existence of common invariant subspaces for
                  semigroups of idempotent operators. It is known that in
                  finite dimensions every such semigroup is simultaneously
                  triangularizable. The question of the existence of even one
                  non-trivial invariant subspace is still open in infinite
                  dimensions.\par Working with semigroups of idempotent
                  operators in Hilbert/Banach vector space settings, we
                  exploit the connection between the purely algebraic
                  structure and the operator structure to show that the
                  answer is affirmative in a number of cases.",
}

@ARTICLE{LR:CJM95,
 author        = "Longstaff, W. E. and Radjavi, H.",
 authauthor    = "Longstaff, W. and Radjavi, H.",
 title         = "On permutability and submultiplicativity of spectral
                  radius",
 journal       = "Canad. J. Math.",
 fjournal      = "Canadian Journal of Mathematics. Journal Canadien de
                  Math{\'e}matiques",
 year          = "1995",
 volume        = "47",
 number        = "5",
 pages         = "1007--1022",
 coden         = "CJMAAB",
 issn          = "0008-414X",
 mrclass       = "47A10 (47A15 47D03)",
 mrnumber      = "1350647 (97d:47005)",
 zblnumber     = "0844.47003",
 doi           = "10.4153/CJM-1995-053-x",
 url           = "https://dx.doi.org/10.4153/CJM-1995-053-x",
 annote        = "Let $r(T)$ denote the spectral radius of the operator $T$,
                  acting on a complex, Hilbert space $H$. Let $\mathcal{S}$
                  be a multiplicative semigroup of operators on $H$. We say
                  that $r$ is \emph{permutable} on $\mathcal{S}$ if $r(ABC) =
                  r(BAC)$, for every $A,B,C \in\mathcal{S}$. We say that $r$
                  is submultiplicative on $\mathcal{S}$ if $r(AE) <
                  r(A)r(B)$, for every $A, B \in\mathcal{S}$. It is known
                  that, if $r$ is permutable on $\mathcal{S}$, then it is
                  submultiplicative. We show that the converse holds in each
                  of the following cases: (i) $\mathcal{S}$ consists of
                  compact operators (ii) $\mathcal{S}$ consists of normal
                  operators (iii) $\mathcal{S}$ is generated by orthogonal
                  projections.",
}

@ARTICLE{LubMit:JACM86,
 author        = "Lubachevsky, Boris and Mitra, Debasis",
 title         = "A chaotic asynchronous algorithm for computing the fixed
                  point of a nonnegative matrix of unit spectral radius",
 journal       = "J. Assoc. Comput. Mach.",
 fjournal      = "Journal of the Association for Computing Machinery",
 year          = "1986",
 volume        = "33",
 number        = "1",
 pages         = "130--150",
 coden         = "JACOAH",
 issn          = "0004-5411",
 mrclass       = "65F15 (15A18 60J10 65W05 68M20)",
 mrnumber      = "820102 (87i:65054)",
 mrreviewer    = "Petr Liebl",
 zblnumber     = "0641.65033",
 zblreviewer   = "H. Matthies",
 doi           = "10.1145/4904.4801",
 url           = "https://dl.acm.org/doi/10.1145/4904.4801",
 annote        = "One of the major problems in parallel computations is the
                  communication between processors and their synchronization.
                  The algorithm for the computation of a positive eigenvector
                  to the unit eigenvalue of a positive, irreducible matrix of
                  unit spectral radius described in this paper makes only
                  mild assumptions with regard to the synchronization of the
                  processors. The number and speed of the processors may vary
                  within certain bounds. Geometric rate of convergence in a
                  projective metric is demonstrated. Examples and counter
                  examples which violate some of the assumptions and are not
                  convergent are included, as well as numerical experiments
                  which show the possible speed-up factor.\par Given a
                  nonnegative, irreducible matrix $P$ of spectral radius
                  unity, there exists a positive vector $\mathbf{r}$ such
                  that $\mathbf{r} = \mathbf{r}P$. If $P$ also happens to be
                  stochastic, then $\mathbf{r}$ gives the stationary
                  distribution of the Markov chain that has state-transition
                  probabilities given by the elements of $P$. This paper
                  gives an algorithm for computing $\mathbf{r}$ that is
                  particularly well suited for parallel processing. The main
                  attraction of our algorithm is that the timing and
                  sequencing restrictions on individual processors are almost
                  entirely eliminated and, consequently, the necessary
                  coordination between processors is negligible and the
                  enforced idle time is also negligible. Under certain mild
                  and easily satisfied restrictions on $P$ and on the
                  implementation of the algorithm, $x(\cdot)$, the vectors of
                  computed values are proved to converge to within a
                  positive, finite constant of proportionality of
                  $\mathbf{r}$. It is also proved that a natural measure of
                  the projective distance of $x(\cdot)$ from $\mathbf{r}$
                  vanishes geometrically fast, and at a rate for which a
                  lower bound is given. We have conducted extensive
                  experiments on random matrices $P$, and the results show
                  that the improvement over the parallel implementation of
                  the synchronous version of the algorithm is substantial,
                  sometimes exceeding the synchronization penalty to which
                  the latter is always subject.",
}

@ARTICLE{LuiOli:AMC14,
 author        = "Lu{\'\i}s, Rafael and Oliveira, Henrique M.",
 authauthor    = "Rafael Lu{\'\i}s and Henrique Oliveira",
 title         = "Products of {$2\times 2$} matrices related to non autonomous
                  {F}ibonacci difference equations",
 journal       = "Appl. Math. Comput.",
 fjournal      = "Applied Mathematics and Computation",
 year          = "2014",
 volume        = "226",
 pages         = "101--116",
 eprinttype    = "arXiv",
 eprint        = "1308.1137",
 issn          = "0096-3003",
 mrclass       = "11B39 (05A15 11B37 39A10)",
 mrnumber      = "3144294",
 zblnumber     = "1354.11012",
 doi           = "10.1016/j.amc.2013.09.075",
 url           = "https://www.sciencedirect.com/science/article/pii/S0096300313010552?via=ihub",
 annote        = "A technique to compute arbitrary products of a class of
                  Fibonacci $2\times2$ square matrices is proved in this
                  work. General explicit solutions for non autonomous
                  Fibonacci difference equations are obtained from these
                  products.\par In the periodic non autonomous Fibonacci
                  difference equations the monodromy matrix, the Floquet
                  multipliers and the Binet's formulas are obtained. In the
                  periodic case explicit solutions are obtained and the
                  solutions are analyzed.",
}

@ARTICLE{Lur:LAA05,
 author        = "Lur, Yung-Yih",
 title         = "On the asymptotic stability of nonnegative matrices in max
                  algebra",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2005",
 volume        = "407",
 pages         = "149--161",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15A48 (15A60 15A99)",
 mrnumber      = "2161921 (2006f:15024)",
 mrreviewer    = "Udrea P{\u{a}}un",
 zblnumber     = "1086.15016",
 zblreviewer   = "Ching-Hua Chang (Hualien)",
 doi           = "10.1016/j.laa.2005.05.017",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379505002867",
 annote        = "In the max algebra system, for $n\times n$ real matrices $A$
                  and $B$, the product $A\otimes B$ has the $(ij)$ entry
                  defined by $\max_{1\le k\le n} a_{ik}b_{kj}$, and for
                  $x\in\mathbb{R}^n$, $A\otimes x$ has the $i$th component
                  defined by $\max_{1\le j\le n}a_{ij}x_{j}$. Fix a norm
                  $\|\cdot\|$ on $\mathbb{R}^n$ and define
                  $\eta(A)=\sup_{\|x\| =1, x\ge 0} \|A\otimes x\|$,
                  $\hat{\eta}(A)=\lim_{k\to\infty} \sup \eta(big\otimes^k
                  A)^{1/k}$. In this paper, the equivalence of the following
                  conditions are proved: (i) $\eta(A)<1$, (ii)
                  $\hat{\eta}(A)<1$, (iii) $\mu(A)<1$; (iv)
                  $\lim_{k\to\infty} \otimes^k A=0$, where $\mu(A)$ is the
                  maximum circuit geometric mean of the directed graph
                  associated with $A$.",
}

@ARTICLE{Lur:LAA06-1,
 author        = "Lur, Yung-Yih",
 title         = "A max version of the generalized spectral radius theorem",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2006",
 volume        = "418",
 number        = "1",
 pages         = "336--346",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15A30 (15A18)",
 mrnumber      = "2257600 (2007g:15017)",
 zblnumber     = "1108.15009",
 zblreviewer   = "C. M. da Fonseca (Coimbra)",
 doi           = "10.1016/j.laa.2006.02.014",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379506000991",
 annote        = "The generalized spectral radius theorem states that the
                  generalized spectral radius and the joint spectral radius
                  of a bounded set of square complex matrices coincide. In
                  this paper a max algebra version of the generalized
                  spectral radius theorem is proposed.\par Let $\Psi$ be a
                  bounded set of $n\times n$ nonnegative matrices in max
                  algebra. In this paper we propose the notions of the max
                  algebra version of the generalized spectral radius
                  $\mu(\Psi)$ of $\Psi$, and the max algebra version of the
                  joint spectral radius $\eta(\Psi)$ of $\Psi$. The max
                  algebra version of the generalized spectral radius theorem
                  $\mu(\Psi) = \eta(\Psi)$ is established. We propose the
                  relationship between the generalized spectral radius
                  $\rho(\Psi)$ of $\Psi$ (in the sense of Daubechies and
                  Lagarias) and its max algebra version $\mu(\Psi)$.
                  Moreover, a generalization of Elsner and van den
                  Driessche's lemma is presented as well.",
}

@ARTICLE{Lur:LAA06-2,
 author        = "Lur, Yung-Yih",
 title         = "A note on a gap result for norms of semigroups of matrices",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2006",
 volume        = "419",
 number        = "2-3",
 pages         = "368--372",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15A18 (20M10)",
 mrnumber      = "2277976 (2007j:15007)",
 zblnumber     = "1112.15016",
 zblreviewer   = "Jaspal Singh Aujla (Jalandhar)",
 doi           = "10.1016/j.laa.2006.05.005",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379506002424",
 annote        = "Let $\|\cdot\|$ be a matrix norm and $\Sigma$ be a bounded
                  set of $n\times n$ complex matrices. For $m\ge 1$, let
                  $\Sigma_m$ be the set of all products of length $m$ of
                  matrices in $\Sigma$. Let $S(\Sigma)$ denote the
                  multiplicative semigroup generated by $\Sigma$. The
                  generalized spectral radius of $\Sigma$, $\rho(\Sigma)$, is
                  defined by $\rho(\Sigma )=\limsup_{m\rightarrow\infty}
                  [\rho_m(\Sigma)]^{\frac{1}{m}}$, where
                  $\rho_m(\Sigma)=\sup_{A\in\Sigma_m}\rho(A)$, and $\rho(A)$
                  is the spectral radius of $A$. The joint spectral radius of
                  $\Sigma$, $\hat{\rho}(\Sigma)$, is defined by
                  $\hat{\rho}(\Sigma )=\limsup_{m\rightarrow\infty}
                  [\hat{\rho}_m(\Sigma,\|\cdot\|)]^{\frac{1}{m}}$, where
                  $\hat{\rho}_m(\Sigma,\|\cdot\|)=\sup_{A\in\Sigma_m}\|A\|$.\par
                  \emph{J.\,P.~Bell}~\cite{Bell:LAA05} proved that there is a
                  gap in the possible growth of
                  $\hat{\rho}_m(\Sigma,\|\cdot\|)$ if $\Sigma$ consists of
                  finitely many $n\times n$ complex matrices. In this note,
                  based on a lemma by Elsner, the author gives an elementary
                  proof of Bell's theorem in bounded case. The author proves
                  that either there is some constant $\alpha>1$ such that
                  $\hat{\rho}_m(\Sigma, \|\cdot\|)> \alpha^m$ for all $m$
                  sufficiently large, or $\hat{\rho}_m(\Sigma,
                  \|\cdot\|)=O(m^{n-1})$. Moreover, $\hat{\rho}_m(\Sigma,
                  \|\cdot\|)=O(m^{n-1})$ if and only if the eigenvalues of
                  every matrix in the semigroup generated by $\Sigma$ are all
                  on or inside the unit circle.",
}

@ARTICLE{LurYang:LAA09,
 author        = "Lur, Yung-Yih and Yang, Wen-Wei",
 title         = "Continuity of the generalized spectral radius in max
                  algebra",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2009",
 volume        = "430",
 number        = "8-9",
 pages         = "2301--2311",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15A18 (15A60 15B48)",
 mrnumber      = "2508296 (2010a:15033)",
 mrreviewer    = "Abbas Salemi",
 zblnumber     = "1168.15005",
 zblreviewer   = "Rodica Covaci (Cluj-Napoca)",
 doi           = "10.1016/j.laa.2008.12.007",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379508005740",
 annote        = "Let $\|\cdot\|$ be an induced matrix norm associated with a
                  monotone norm on $\mathbb{R}^n$ and $\beta$ be the
                  collection of all nonempty closed and bounded subsets of
                  $n\times n$ nonnegative matrices under this matrix norm.
                  For $\Psi,\Phi\in \beta$, the Hausdorff metric $H$ between
                  $\Psi$ and $\Phi$ is given by $H(\Psi,\Phi) =
                  \max\{\sup_{A\in\Psi}\inf_{B\in\Phi}\|A-B\|,~
                  \sup_{B\in\Phi}\inf_{A\in\Psi}\|A-B\|\}$. The max algebra
                  system consists of the set of nonnegative numbers with sum
                  $a\otimes b = \max\{a,b\}$ and the standard product $ab$
                  for $a,b\ge 0$. For $n\times n$ nonnegative matrices $A,B$
                  their product is denoted by $A\otimes B$, where $[A\otimes
                  B]_{ij} = \max_{1\le k\le n}a_{ik}b_{kj}$. For each
                  $\Psi\in\beta$, the max algebra version of the generalized
                  spectral radius of is $\mu(\Psi) = \limsup_{m\to\infty}
                  [\sup_{A\in\Psi^{m}_{\otimes}}\mu(A)]^{1/m}$, where
                  $\Psi^{m}_{\otimes}= \{A_1\otimes A_2\otimes\cdots\otimes
                  A_m : A_i\in\Psi\}$. Here $\mu(A)$ is the maximum circuit
                  geometric mean. In this paper we prove that the max algebra
                  version of the generalized spectral radius is continuous on
                  the Hausdorff metric space $(\beta,H)$. The notion of the
                  max algebra version of simultaneous nilpotence of matrices
                  is also proposed. Necessary and sufficient conditions for
                  the max algebra version of simultaneous nilpotence of
                  matrices are presented as well.",
}

@ARTICLE{Maesumi:CMA00,
 author        = "Maesumi, M.",
 title         = "Joint spectral radius and {H}{\"o}lder regularity of
                  wavelets",
 journal       = "Comput. Math. Appl.",
 fjournal      = "Computers {\&} Mathematics with Applications. An
                  International Journal",
 year          = "2000",
 volume        = "40",
 number        = "1",
 pages         = "145--155",
 coden         = "CMAPDK",
 issn          = "0898-1221",
 mrclass       = "42C40",
 mrnumber      = "1768971",
 zblnumber     = "0948.42019",
 doi           = "10.1016/S0898-1221(00)00148-6",
 url           = "https://www.sciencedirect.com/science/article/pii/S0898122100001486",
 note          = "{A}pproximation in mathematics (Memphis, TN, 1997)",
 annote        = "We give preliminary results on the H{\"o}lder exponent of
                  wavelets of compact support. In particular, we give a
                  nearly complete map of this exponent for the family of
                  four-coefficient multiresolution analyses and determine the
                  smoothest one. This will resolve two conjectures by Colella
                  and Heil. Wavelets of compact support can be generated via
                  infinite products of certain matrices. The rate of growth
                  of these products determines the regularity of the wavelet.
                  This rate can be determined via joint, generalized, or
                  common spectral radius of the given set of matrices. We
                  outline a method for calculating this radius for a given
                  set of matrices. The method relies on guessing the
                  particular finite optimal product which satisfies the
                  finiteness conjecture and exhibits the fastest growth.
                  Then, we generate an optimal unit ball by taking the convex
                  hull of the action of the semigroup of matrices, scaled by
                  their joint radius, on the invariant ball of the scaled
                  optimal product. If this process terminates in a finite
                  number of steps and the convex hull does not grow, then the
                  guessed optimal product is confirmed and the joint radius
                  is determined.",
}

@INCOLLECTION{Maesumi:TIC95,
 author        = "Maesumi, Mohsen",
 title         = "Optimum unit ball for joint spectral radius: an example from
                  four-coefficient {MRA}",
 booktitle     = "Approximation theory {VIII}, {V}ol.\ 2 ({C}ollege {S}tation,
                  {TX}, 1995)",
 series        = "Ser. Approx. Decompos.",
 publisher     = "World Sci. Publ., River Edge, NJ",
 year          = "1995",
 volume        = "6",
 pages         = "267--274",
 mrclass       = "15A60 (42C15)",
 mrnumber      = "1471793 (98f:15019)",
 zblnumber     = "0927.42025",
 url           = "https://www.math.lamar.edu/faculty/maesumi/papers/Optimum.pdf",
 annote        = "We give the exact value of joint spectral radius for certain
                  matrices by explicitly constructing an optimum unit ball
                  and an operator norm. Our example matrices are associated
                  with 4-coefficient dilation equations which generate a
                  multiresolution analysis. The method proposed here can be
                  used in the search for orthogonal solution of the dilation
                  equation with the highest H{\"o}lder exponent. It can also
                  be used to decide if a given product of matrices is
                  optimal, i.e., it satisfies the finiteness conjecture. The
                  optimal ball is generated as the convex hull of action of
                  semigroup of matrices, normalized by their joint spectral
                  radius, on extreme points of a set of unit diameter which
                  is maximal, convex, symmetric, compact and invariant under
                  the optimal product.",
}

@ARTICLE{Maesumi:LAA96,
 author        = "Maesumi, Mohsen",
 title         = "An efficient lower bound for the generalized spectral radius
                  of a set of matrices",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "1996",
 volume        = "240",
 pages         = "1--7",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15A60 (42C15 47A10)",
 mrnumber      = "1387282 (97b:15027)",
 zblnumber     = "0848.15017",
 zblreviewer   = "F.Uhlig (Auburn)",
 doi           = "10.1016/0024-3795(94)00171-5",
 url           = "https://www.sciencedirect.com/science/article/pii/0024379594001715",
 annote        = "For a finite collection $\Sigma$ of $q\times q$ matrices,
                  its, generalized spectral radius $\rho(\Sigma)$ is defined
                  as the lim sup of the maximal $n^{\text{th}}$ root of the
                  spectral radius of words of length $n$ from $\Sigma$ as
                  $n\to\infty$. This concept has applications to wavelets.
                  For fixed $n$, the naive computation of $\rho_n (\Sigma)=
                  \max_{A\in\Sigma^n} |\rho(A)|^{\frac{1}{n}}$ takes $|\Sigma
                  |^n$ spectral radii evaluations to find the maximum. This
                  paper shortens the process to $|\Sigma|^n/n$ evaluations by
                  observing that cyclically rearranged words from $\Sigma^n$
                  have the same characteristic polynomial, hence spectral
                  radius, by eliminating products that are a power of shorter
                  products and by using the Dedekind--Liouville transforms
                  from arithmetic number theory on the MacMahon formula for
                  counting cyclically different words of length $n$.",
}

@INCOLLECTION{Maesumi:AT98,
 author        = "Maesumi, Mohsen",
 title         = "Calculating joint spectral radius of matrices and
                  {H}{\"o}lder exponent of wavelets",
 booktitle     = "Approximation theory {IX}, {V}ol. 2 ({N}ashville, {TN},
                  1998)",
 series        = "Innov. Appl. Math.",
 publisher     = "Vanderbilt Univ. Press",
 address       = "Nashville, TN",
 year          = "1998",
 pages         = "205--212",
 mrclass       = "42C40 (65F15)",
 mrnumber      = "1744409",
 zblnumber     = "0916.42026",
 annote        = "The joint spectral radius (jsr) of a bounded collection of
                  matrices $\mathcal{M}$ is the smallest positive number $r$
                  such that $\mathcal{M}/r$ generates a norm-bounded
                  semigroup. This quantity can be used to determine
                  H{\"o}lder regularity of compactly supported wavelets. The
                  finiteness conjecture of Lagarias and Daubechies states
                  that if $\mathcal{M}$ is finite then a certain optimal
                  product $P$ of $n$ elements of $\mathcal{M}$ attains the
                  maximal growth rate, $\rho(P)=\text{jsr}(\mathcal{M})^n$.
                  We describe an algorithm which is conjectured to terminate
                  iff $P$ is an optimal product. Experimental results of the
                  algorithm relating to the H{\"o}lder exponent of
                  four-coefficient multiresolution analyses are presented.",
}

@INCOLLECTION{Maesumi:LNPAM00,
 author        = "Maesumi, Mohsen",
 title         = "Calculating the spectral radius of a set of matrices",
 booktitle     = "Wavelet analysis and multiresolution methods
                  ({U}rbana-{C}hampaign, {IL}, 1999)",
 series        = "Lecture Notes in Pure and Appl. Math.",
 publisher     = "Dekker",
 address       = "New York",
 year          = "2000",
 volume        = "212",
 pages         = "255--272",
 mrclass       = "65F15 (65T60)",
 mrnumber      = "1777996 (2001e:65063)",
 zblnumber     = "0965.42021",
 zblreviewer   = "Rodica Covaci (Cluj-Napoca)",
 annote        = "The paper presents an algorithm for calculating the joint or
                  generalized spectral radius of a set of matrices. The
                  algorithm is conjectured to verify, in a finite number of
                  steps, if a certain product is the optimal product
                  satisfying the finiteness conjecture of Daubechies and
                  Lagarias. The algorithm has been applied to matrices
                  arising from the regularity analysis of multiresolution
                  analyses and compactly supported wavelets. In particular,
                  four- and six-coefficient MRA's with the highest H{\"o}lder
                  exponent are presented.",
}

@INPROCEEDINGS{Maesumi:CDC05,
 author        = "Maesumi, Mohsen",
 title         = "Construction of Optimal Norms for Semi-Groups of Matrices",
 booktitle     = "Proceedings of the 44th {IEEE} Conference on Decision and
                  Control and European Control Conference 2005, Seville,
                  Spain, December 12--15",
 year          = "2005",
 pages         = "3013--3018",
 doi           = "10.1109/CDC.2005.1582623",
 annote        = "The notion of spectral radius of a set of matrices is a
                  natural extension of spectral radius of a single matrix.
                  The Finiteness Conjecture (FC) claims that among the
                  infinite products made from the elements of a given finite
                  set of matrices, there is a certain periodic product, made
                  from the repetition of a finite product (the optimal
                  product), whose rate of growth is maximal. FC has been
                  disproved. In this paper it is conjectured that FC is
                  almost always true, and an algorithm is presented to verify
                  the optimality of a given product. The algorithm uses
                  optimal norms, as a special subset of extremal norms. The
                  algorithm has successfully calculated the spectral radius
                  of the pair of matrices associated with compactly supported
                  multi-resolution analyses and wavelets.",
}

@ARTICLE{Maesumi:LAA08,
 author        = "Maesumi, Mohsen",
 title         = "Optimal norms and the computation of joint spectral radius
                  of matrices",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2008",
 volume        = "428",
 number        = "10",
 pages         = "2324--2338",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15A60 (47A13)",
 mrnumber      = "2408030 (2009a:15087)",
 mrreviewer    = "Morteza Seddighin",
 zblnumber     = "1138.65030",
 doi           = "10.1016/j.laa.2007.09.036",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379507004521",
 annote        = "The notion of spectral radius of a set of matrices is a
                  natural extension of spectral radius of a single matrix.
                  The finiteness conjecture (FC) claims that among the
                  infinite products made from the elements of a given finite
                  set of matrices, there is a certain periodic product, made
                  from the repetition of the optimal product, whose rate of
                  growth is maximal. FC has been disproved.\par In this paper
                  it is conjectured that FC is almost always true, and an
                  algorithm is presented to verify the optimality of a given
                  product. The algorithm uses optimal norms, as a special
                  subset of extremal norms. Several conjectures related to
                  optimal norms and non-decomposable sets of matrices are
                  presented. The algorithm has successfully calculated the
                  spectral radius of several parametric families of pairs of
                  matrices associated with compactly supported
                  multi-resolution analyses and wavelets. The results of
                  related numerical experiments are presented.",
}

@ARTICLE{ManSim:TCS78,
 author        = "Mandel, Arnaldo and Simon, Imre",
 title         = "On finite semigroups of matrices",
 journal       = "Theoret. Comput. Sci.",
 fjournal      = "Theoretical Computer Science",
 year          = "1977/78",
 volume        = "5",
 number        = "2",
 pages         = "101--111",
 issn          = "0304-3975",
 mrclass       = "20M20 (20M35)",
 mrnumber      = "473070",
 mrreviewer    = "W. Wechler",
 zblnumber     = "0368.20049",
 doi           = "10.1016/0304-3975(77)90001-9",
 url           = "https://www.sciencedirect.com/science/article/pii/0304397577900019",
 annote        = "Finite semigroups of $n$ by $n$ matrices over the naturals
                  are characterized both by algebraic and combinatorial
                  methods. Next we show that the cardinality of a finite
                  semigroup $S$ of $n$ by $n$ matrices over a field is
                  bounded by a function depending only on $n$, the number of
                  generators of $S$ and the maximum cardinality of its
                  subgroups. As a consequence, given $n$ and $k$, there
                  exist, up to isomorphism, only a finite number of finite
                  semigroups of n by n matrices over the rationals, generated
                  by at most $k$ elements. Among other applications to
                  Automaton Theory, we show that it is decidable whether the
                  behavior of a given $N-\Sigma$ automaton is bounded.",
}

@ARTICLE{MasWir:LAA14,
 author        = "Mason, Oliver and Wirth, Fabian",
 title         = "Extremal norms for positive linear inclusions",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2014",
 volume        = "444",
 pages         = "100--113",
 eprinttype    = "arXiv",
 eprint        = "1306.3814",
 issn          = "0024-3795",
 mrclass       = "34A60 (15A30 15A60 52A21)",
 mrnumber      = "3145832",
 zblnumber     = "1285.15012",
 doi           = "10.1016/j.laa.2013.11.020",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379513007180",
 annote        = "For finite-dimensional linear semigroups which leave a
                  proper cone invariant it is shown that irreducibility with
                  respect to the cone implies the existence of an extremal
                  norm. In case the cone is simplicial a similar statement
                  applies to absolute norms. The semigroups under
                  consideration may be generated by discrete-time systems,
                  continuous-time systems or continuous-time systems with
                  jumps. The existence of extremal norms is used to extend
                  results on the Lipschitz continuity of the joint spectral
                  radius beyond the known case of semigroups that are
                  irreducible in the representation theory interpretation of
                  the word.",
}

@ARTICLE{Mate:PAMS98,
 author        = "Mate, L{\'a}szl{\'o}",
 authauthor    = "M{\'a}t{\'e}, L{\'a}szl{\'o}",
 title         = "On the infinite product of operators in {H}ilbert space",
 journal       = "Proc. Amer. Math. Soc.",
 fjournal      = "Proceedings of the American Mathematical Society",
 year          = "1998",
 volume        = "126",
 number        = "2",
 pages         = "535--543",
 issn          = "0002-9939",
 mrclass       = "47A05",
 mrnumber      = "1415333",
 mrreviewer    = "Volker Wrobel",
 zblnumber     = "0893.47001",
 doi           = "10.1090/S0002-9939-98-04067-2",
 url           = "https://www.ams.org/journals/proc/1998-126-02/S0002-9939-98-04067-2/",
 annote        = "We give a necessary and sufficient condition for a certain
                  set of infinite products of linear operators to be zero. We
                  shall investigate also the case when this set of infinite
                  products converges to a non-zero operator. The main device
                  in these results is a weighted version of the K{\"o}nig
                  Lemma for infinite trees in graph theory.",
}

@ARTICLE{Mate:FM99,
 author        = "M{\'a}t{\'e}, L{\'a}szl{\'o}",
 title         = "On infinite composition of affine mappings",
 journal       = "Fund. Math.",
 fjournal      = "Fundamenta Mathematicae",
 year          = "1999",
 volume        = "159",
 number        = "1",
 pages         = "85--90",
 issn          = "0016-2736",
 mrclass       = "28D99 (54H20)",
 mrnumber      = "1669710",
 mrreviewer    = "Irene Hueter",
 zblnumber     = "0939.47006",
 zblreviewer   = "Aleksandr A. Mekler (Dresden)",
 url           = "https://eudml.org/doc/212321",
 annote        = "Let $\{F_{i} : i = 1,\ldots,N\}$ be affine mappings of
                  $\mathbb{R}^{n}$. It is well known that if there exists $j
                  \leq 1$ such that for every $\sigma_{1},\ldots,\sigma_{j}
                  \in \{1,\ldots,N\}$ the composition
                  $F_{\sigma_{1}}\circ\cdots\circ F_{\sigma_{j}}$ is a
                  contraction, then for any infinite sequence $\sigma_{1},
                  \sigma_{2},\ldots\in \{1,\ldots,N\}$ and any
                  $z\in\mathbb{R}^{n}$, the sequence
                  $F_{\sigma_{1}}\circ\cdots\circ F_{\sigma_{n}}(z)$ is
                  convergent and the limit is independent of $z$. We prove
                  the following converse result:\par \textbf{Theorem.}
                  \emph{If $F_{\sigma_{1}}\circ\cdots\circ F_{\sigma_{n}}(z)$
                  is convergent for any $z\in\mathbb{R}^{n}$ and any $\sigma
                  = \{\sigma_{1}, \sigma_{2},\ldots\}$ belonging to some
                  subshift $\Sigma$ of $N$ symbols (and the limit is
                  independent of $z$), then there exists $j \geq 1$ such that
                  for every $\sigma = \{\sigma_{1}, \sigma_{2},\ldots\} \in
                  \Sigma$ the composition $F_{\sigma_{1}}\circ\cdots\circ
                  F_{\sigma_{j}}$ is a contraction.}\par This result can be
                  considered as a generalization of the main theorem of
                  \emph{I.~Daubechies} and
                  \emph{J.\,C.~Lagarias}~\cite[p.~239]{DaubLag:LAA92}. The
                  proof involves some easy but non-trivial combinatorial
                  considerations. The most important tool is a weighted
                  version of the K{\"o}nig Lemma for infinite trees in graph
                  theory.",
}

@PHDTHESIS{Mejstrik:19,
 author        = "Mejstrik, Thomas",
 title         = "Joint spectral radius and subdivision schemes",
 school        = "Fakult{\"a}t f{\"u}r Mathematik",
 address       = "Universit{\"a}t Wien, Besprechungszimmer 09. Stock,
                  Oskar-Morgenstern-Platz 1",
 year          = "2019",
 pagetotal     = "113",
 url           = "http://www.tommsch.com/misc/diss.pdf",
 annote        = "This thesis extends the matrix based approach to the setting
                  of multiple subdivision schemes studied in [Sauer, 2012].
                  Multiple subdivision schemes, in contrast to stationary and
                  nonstationary schemes, allow for level dependent
                  subdivision weights and for level dependent choice of the
                  dilation matrices. The latter property of multiple
                  subdivision makes the standard definition of the transition
                  matrices, crucial ingredient of the matrix approach in the
                  stationary and non-stationary settings, inapplicable. We
                  show how to avoid this obstacle and characterize the
                  convergence of multiple subdivision schemes in terms of the
                  joint spectral radius of certain square matrices derived
                  from subdivision weights.\par Albeit the characterization
                  of the convergence of multiple subdivision schemes in terms
                  of the joint spectral radius is elegant, the numerical
                  computation of the joint spectral radius still poses big
                  problems. In several papers of 2013~--~2016, Guglielmi and
                  Protasov made a breakthrough in the problem of the joint
                  spectral radius computation, developing the invariant
                  polytope algorithm, which for most matrix families finds
                  the exact value of the joint spectral radius. This
                  algorithm found many applications in problems of functional
                  analysis, approximation theory, combinatorics, etc. In this
                  thesis we propose a modification of the invariant polytope
                  algorithm making it roughly three times faster and suitable
                  for higher dimensions. The modified version works for most
                  matrix families of dimensions up to 25, for non-negative
                  matrices the dimension is up to 3000.\par Besides, we
                  introduce a new, fast algorithm for computing good lower
                  bounds for the joint spectral radius, which finds in most
                  cases the exact value of the joint spectral radius in less
                  than 5 seconds. Corresponding examples and statistics of
                  numerical results are provided.",
}

@ARTICLE{Mejstrik:ACMTMS20,
 author        = "Mejstrik, Thomas",
 title         = "Algorithm 1011: {I}mproved invariant polytope algorithm and
                  applications",
 journal       = "ACM Trans. Math. Software",
 fjournal      = "Association for Computing Machinery. Transactions on
                  Mathematical Software",
 year          = "2020",
 volume        = "46",
 number        = "3",
 pages         = "Art. 29, 26",
 eprinttype    = "arXiv",
 eprint        = "1812.03080",
 issn          = "0098-3500",
 mrclass       = "65F99 (15A60)",
 mrnumber      = "4161245",
 zblnumber     = "1484.65090",
 doi           = "10.1145/3408891",
 url           = "https://dl.acm.org/doi/10.1145/3408891",
 annote        = "In several papers of 2013~--~2016, Guglielmi and Protasov
                  made a breakthrough in the problem of the joint spectral
                  radius computation, developing the invariant polytope
                  algorithm which for most matrix families finds the exact
                  value of the joint spectral radius. This algorithm found
                  many applications in problems of functional analysis,
                  approximation theory, combinatorics, etc.. In this paper we
                  propose a modification of the invariant polytope algorithm
                  making it roughly 3 times faster and suitable for higher
                  dimensions. The modified version works for most matrix
                  families of dimensions up to 25, for non-negative matrices
                  the dimension is up to three thousand. Besides we introduce
                  a new, fast algorithm for computing good lower bounds for
                  the joint spectral radius. The corresponding examples and
                  statistics of numerical results are provided. Several
                  applications of our algorithms are presented. In
                  particular, we find the exact values of the regularity
                  exponents of Daubechies wavelets of high orders and the
                  capacities of codes that avoid certain difference
                  patterns.",
}

@ARTICLE{Mejstrik:DRNA22,
 author        = "Mejstrik, Thomas",
 title         = "The finiteness conjecture for {$3\times 3$} binary
                  matrices",
 journal       = "Dolomites Research Notes on Approximation",
 fjournal      = "Dolomites Research Notes on Approximation",
 year          = "2022",
 volume        = "15",
 number        = "5",
 pages         = "24--38",
 issn          = "2035-6803",
 mrclass       = "15A18 (65F10)",
 mrnumber      = "4538598",
 url           = "https://drna.padovauniversitypress.it/2022/5/3",
 doi           = "10.14658/pupj-drna-2022-5-3",
 annote        = "The invariant polytope algorithm was a breakthrough in the
                  joint spectral radius computation, allowing to find the
                  exact value of the joint spectral radius for most matrix
                  families \cite{GugProt:FCM13,GugProt:SIAMJMAA16}. This
                  algorithm found many applications in problems of functional
                  analysis, approximation theory, combinatorics, etc. In this
                  paper we propose a modification of the invariant polytope
                  algorithm enlarging the class of problems to which it is
                  applicable. Precisely, we introduce mixed numeric and
                  symbolic computations. A further minor modification of
                  augmenting the input set with additional matrices speeds up
                  the algorithm in certain cases. With this modifications we
                  are able to automatically prove the finiteness conjecture
                  for all pairs of binary ${3\times 3}$ matrices and sign
                  ${2\times 2}$ matrices.",
}

@ARTICLE{MejProt:Cacolo23,
 author        = "Mejstrik, Thomas and Protasov, Vladimir {\relax{}Yu}.",
 authauthor    = "Mejstrik, Thomas and Protasov, Vladimir",
 title         = "Elliptic polytopes and invariant norms of linear operators",
 journal       = "Calcolo",
 fjournal      = "Calcolo. A Quarterly on Numerical Analysis and Theory of
                  Computation",
 volume        = "60",
 year          = "2023",
 number        = "4",
 pages         = "Paper No. 56",
 issn          = "0008-0624,1126-5434",
 mrclass       = "52A21 (15A60 39A30 90C90)",
 mrnumber      = "4668871",
 zblnumber     = "07783083",
 eprinttype    = "arXiv",
 eprint        = "2107.02610",
 doi           = "10.1007/s10092-023-00547-z",
 annote        = "We address the problem of constructing elliptic polytopes
                  in~${\mathbb R}^d$, which are convex hulls of finitely many
                  two-dimensional ellipses with a common center. Such sets
                  arise in the study of spectral properties of matrices,
                  asymptotics of long matrix products, in the Lyapunov
                  stability, etc. The main issue in the construction is to
                  decide whether a given ellipse is in the convex hull of
                  others. The computational complexity of this problem is
                  analysed by considering an equivalent optimisation problem.
                  We show that the number of local extrema of that problem
                  may grow exponentially in~$d$. For $d=2, 3$, it admits an
                  explicit solution for an arbitrary number of ellipses; for
                  higher dimensions, several geometric methods for
                  approximate solutions are derived. Those methods are
                  analysed numerically and their efficiency is demonstrated
                  in applications.",
}

@MISC{MejReif:ArXiv23,
 author        = "Mejstrik, Thomas and Reif, Ulrich",
 title         = "Hybrid approach to the joint spectral radius computation",
 howpublished  = "ArXiv.org e-Print archive",
 year          = "2023",
 month         = aug,
 eprinttype    = "arXiv",
 eprint        = "2308.05244",
 doi           = "10.48550/arXiv.2308.05244",
 annote        = "The problem of computing the \emph{joint spectral radius}
                  (JSR) is hard, although in recent times sophisticated
                  algorithms to compute it were devised~\cite{AJPR:SIAMJCO14,
                  GugProt:FCM13, ParJdb:LAA08, MolReif:LAA14}. Though, among
                  them are only two which compute the JSR exactly and work
                  for a large class of matrix families -- the \emph{invariant
                  polytope algorithm} (ipa)~\cite{GugProt:FCM13,
                  GugProt:SIAMJMAA16, GWZ:SIAMJMA05, GugZen:JCA07,
                  GugZen:SIAMJMAA09, Mejstrik:ACMTMS20, MejProt:Cacolo23}
                  and the \emph{finite expressible tree algorithm} (feta) by
                  M{\"o}ller and Reif~\cite{Moeller15, MolReif:LAA14}. We
                  will discuss the ideas behind those algorithms after
                  defining the crucial concepts. Afterwards we make the
                  attempt of combining them.\par In this paper we propose a
                  modification to the invariant polytope algorithm (ipa)
                  using ideas of the finite expressible tree algorithm
                  (feta). We show that our new feta-flavoured-ipa applies to
                  a wider range of matrix families.",
}

@ARTICLE{Miellow:CRACP74,
 author        = "Miellou, Jean-Claude",
 title         = "It{\'e}rations chaotiques {\`a} retards",
 journal       = "C. R. Acad. Sci. Paris S{\'e}r. A",
 fjournal      = "Comptes Rendus Hebdomadaires des S{\'e}ances de
                  l'Acad{\'e}mie des Sciences. S{\'e}rie A. Sciences
                  Math{\'e}matiques",
 year          = "1974",
 volume        = "278",
 pages         = "957--960",
 issn          = "0302-8429",
 mrclass       = "65H10 (47H10)",
 mrnumber      = "362887",
 mrreviewer    = "L. Collatz",
 zblnumber     = "0314.65028",
 language      = "english",
 annote        = "\emph{D.~Chazan} and \emph{W.~Miranker}~\cite{ChM:LAA69}
                  proved that the chaotic iteration for the linear equation
                  $u=Au+r$ converges if the spectral radius $\rho(|A|)<1$.
                  Here this result is generalized to nonlinear equations $u =
                  F(u)$, where $F:D(F)\subset E\to E$ and a
                  $E=\prod_{i=1}^{\alpha}E_i$ is the product of Banach spaces
                  $E_i$, with norms $\|~\|_i$ $(i = 1,\ldots,\alpha)$. Taking
                  on $E$ the vectorial norm $\|(u_1,\ldots,u_\alpha)\| =
                  \left(\|u_1\|_1,\ldots,\|u_\alpha\|_\alpha\right)$ it is
                  assumed that there is a nonnegative
                  $\alpha\times\alpha$-matrix $T$, $\rho(T)<1$, such that for
                  a fixed point $u^*$ of $F$ in $D(F)$ and all $v\in D(F)$
                  $\|F(u^*) - F(v)\| \le T(\|u^* - v\|)$ holds. Then it is
                  shown that the chaotic iteration for $u=F(u)$, defined as
                  in the paper of Chazan--Miranker, converges to $u^*$ for
                  any starting vector near $u^*$. Bounds for the convergence
                  rate are given.",
}

@ARTICLE{Miellow:CRACP75,
 author        = "Miellou, Jean-Claude",
 title         = "It{\'e}rations chaotiques {\`a} retards; {\'e}tudes de la
                  convergence dans le cas d'espaces partiellement
                  ordonn{\'e}s",
 journal       = "C. R. Acad. Sci. Paris S{\'e}r. A-B",
 fjournal      = "Comptes Rendus Hebdomadaires des S{\'e}ances de
                  l'Acad{\'e}mie des Sciences. S{\'e}ries A et B",
 year          = "1975",
 volume        = "280",
 pages         = "A233--A236",
 issn          = "0151-0509",
 mrclass       = "65J05",
 mrnumber      = "391499",
 mrreviewer    = "L. Collatz",
 zblnumber     = "0332.65060",
 language      = "english",
 annote        = "Here is given a general result of convergence in the case of
                  partially ordered spaces for certain algorithms, which one
                  applies to a problem of quasi-variational inequalities.",
}

@PHDTHESIS{Moeller15,
 author        = "M{\"o}ller, Claudia",
 title         = "A New Strategy for Exact Determination of the Joint Spectral
                  Radius",
 school        = "Fachbereich Mathematik",
 address       = "Technische Universit{\"a}t Darmstadt, Germany",
 year          = "2015",
 url           = "https://tuprints.ulb.tu-darmstadt.de/4603/",
 annote        = "Computing the joint spectral radius of a finite matrix
                  family is, though interesting for many applications, a
                  difficult problem. This work proposes a method for
                  determining the exact value which is based on
                  graph-theoretical ideas. In contrast to some other
                  algorithms in the literature, the purpose of the approach
                  is not to find an extremal norm for the matrix family. To
                  validate that the finiteness property (FP) is satisfied for
                  a certain matrix product, a tree is to be analyzed whose
                  nodes code sets of matrix products. A sufficient, and in
                  certain situations also necessary, criterion is given by
                  existence of a finite tree with special properties, and an
                  algorithm for searching such a tree is proposed. The
                  suggested method applies in case of several FP-products as
                  well and is not limited to asymptotically simple matrix
                  families. In the smoothness analysis of subdivision
                  schemes, joint spectral radius determination is crucial to
                  detect H{\"o}lder regularity. The palindromic symmetry of
                  matrices, which results from symmetric binary subdivision,
                  is considered in the context of set-valued trees. Several
                  illustrating examples explore the capabilities of the
                  approach, consolidated by examples from subdivision.",
}

@ARTICLE{MolReif:LAA14,
 author        = "M{\"o}ller, Claudia and Reif, Ulrich",
 title         = "A tree-based approach to joint spectral radius
                  determination",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2014",
 volume        = "463",
 pages         = "154--170",
 month         = dec,
 issn          = "0024-3795",
 mrclass       = "15A60",
 mrnumber      = "3262394",
 zblnumber     = "1300.15004",
 doi           = "10.1016/j.laa.2014.08.009",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379514005412",
 annote        = "We suggest a novel method to determine the joint spectral
                  radius of finite sets of matrices by validating the
                  finiteness property. It is based on finding a certain
                  finite tree with nodes representing sets of matrix
                  products. Our approach accounts for cases where one or
                  several matrix products satisfy the finiteness property.
                  Moreover, is potentially functional even for reducible sets
                  of matrices.",
}

@ARTICLE{Moss:ACM10,
 author        = "M{\"o}{\ss}ner, Bernhard",
 title         = "On the joint spectral radius of matrices of order 2 with
                  equal spectral radius",
 journal       = "Adv. Comput. Math.",
 fjournal      = "Advances in Computational Mathematics",
 year          = "2010",
 volume        = "33",
 number        = "2",
 pages         = "243--254",
 issn          = "1019-7168",
 mrclass       = "15A18 (15A60)",
 mrnumber      = "2659589 (2011h:15018)",
 mrreviewer    = "Line Baribeau",
 zblnumber     = "1197.15007",
 zblreviewer   = "Alessandra Bernardi (Povo)",
 doi           = "10.1007/s10444-009-9130-y",
 url           = "https://link.springer.com/article/10.1007/s10444-009-9130-y",
 annote        = "Many applications like stability analysis of stochastic
                  dynamical systems, or in approximation theory use the joint
                  spectral radius to verify the convergence or smoothness of
                  subdivision algorithms. The joint spectral radius of two
                  matrices $A_1,A_2 \in\mathbb{R}^{d\times d}$ is
                  \[\text{jsr}(A_1,A_2 ):=\lim_{n\to\infty} \max\{\|T_1
                  T_2\cdots T_n\|^{1/n}~ |~ T_i = A_1\text{ or }A_2\}\] and
                  it measures the growth (or decrease) rate of products of
                  matrices taken from a set.\par In the Introduction of this
                  paper, an accurate presentation of the problem of computing
                  spectral radius and of the literature around this subject
                  is given. The author shows the difficulty of determining
                  the joint spectral radius and the few results that were
                  produced in this direction starting from 1990 by
                  \emph{V.~Kozyakin}~\cite{Koz:AiT90:6:e} that proved that
                  $\left\{(A,B)\in\left(\mathbb{R}^{2\times 2}\right)^2~ |~
                  \text{jsr} (A,B)<1\right\}$ is not a semi-algebraic set.
                  Some of the known results are about the polynomial-time
                  approximation algorithms for the joint spectral radius
                  (\emph{V.\,D.~Blondel} and \emph{J.\,N.~Tsitsiklis} showed
                  in 1997 that until P=NP there is no such an approximation
                  algorithm~\cite{TB:MCSS97-1}. The joint spectral radius
                  seems to be known in the case of pairs of matrices of order
                  $2$ both when one has negative determinant
                  (\emph{M.~Br{\"o}ker} and
                  \emph{X.~Zhou}~\cite{BZh:SAIMJMAA00}) and when of the
                  matrix is singular (\emph{R.~Lima} and
                  \emph{M.~Rahibe}~\cite{LimaRah:JPA94}). Other examples were
                  given by \emph{T.\,N.\,T.~Goodman}, \emph{C.\,A.~Micchelli}
                  and \emph{J.\,D.~Ward} [in: Recent advances in wavelet
                  analysis. Boston, MA: Academic Press, Inc. Wavelet Anal.
                  Appl. 3, 335--360 (1994)] and by \emph{L.~Villemoes} [SIAM
                  J. Math. Anal. 25, No. 5, 1433--1460 (1994)].\par The main
                  result of the paper is the computation of two palindromic
                  matrices $L,P$ of order $2$. It is proved that
                  $\text{jsr}(L,R)=
                  \max\left(\rho(L),\sqrt{\rho(L,R)}\right)$ where $\rho(A)$
                  is the spectral radius of the matrix $A$:
                  $\rho(A):=\max\{|\lambda|: \lambda\text{ is an eigenvalue
                  of }A\}$.\par Moreover, the technique of the proof of this
                  result is used by the author to generalize the computation
                  of the spectral radius the similar matrices with certain
                  relation between the entries.\par At the end of the paper,
                  some examples and an application of the joint spectral
                  radius formula on a subdivision scheme is given.",
}

@ARTICLE{Moh:JSP22,
 author        = "Mohammadpour, Reza",
 title         = "{L}yapunov spectrum properties and continuity of the lower
                  joint spectral radius",
 journal       = "J. Stat. Phys.",
 fjournal      = "Journal of Statistical Physics",
 year          = "2022",
 volume        = "187",
 number        = "3",
 pages         = "Paper No. 23, 29",
 eprinttype    = "arXiv",
 eprint        = "2001.03958",
 issn          = "0022-4715",
 mrclass       = "37D35 (37A60 37D30 37H15 37N40)",
 mrnumber      = "4408461",
 zblnumber     = "07510775",
 doi           = "10.1007/s10955-022-02910-w",
 url           = "https://link.springer.com/article/10.1007/s10955-022-02910-w",
 annote        = "In this paper we study ergodic optimization and multifractal
                  behavior of Lyapunov exponents for matrix cocycles. We show
                  that the restricted variational principle holds for generic
                  cocycles (in the sense of \emph{C.~Bonatti,
                  X.~G{\'o}mez-Mont, and M.~Viana.} G{\'e}n{\'e}ricit{\'e}
                  d{\'e}xposants de Lyapunov nonnuls pour des produits
                  d{\'e}terministes de matrices. Annales de l'Institut Henri
                  Poincare (C) Non Linear Analysis, vol. 20, Elsevier, 2003,
                  pp. 579--624.) over mixing subshifts of finite type. We
                  also show that the Lyapunov spectrum is equal to the
                  closure of the set where the entropy spectrum is positive
                  for such cocycles. Moreover, we show the continuity of the
                  entropy spectrum at boundary of Lyapunov spectrum in the
                  sense that $h_{top}(E(\alpha_{t}))\ \rightarrow
                  h_{top}(E(\beta(\mathcal{A}))$, where $E(\alpha)=\{x\in X:
                  \lim_{n\rightarrow \infty}\frac{1}{n}\log
                  \|\mathcal{A}^{n}(x)\|=\alpha\}$, for such cocycles. \par
                  We prove the continuity of the lower joint spectral radius
                  for linear cocycles under the assumption that linear
                  cocycles satisfy a cone condition.",
}

@INPROCEEDINGS{MOS:GTS99,
 author        = "Moision, Bruce E. and Orlitsky, Alon and Siegel, Paul H.",
 authauthor    = "Moision, Bruce and Orlitsky, Alon and Siegel, Paul",
 title         = "Bounds on the rate of codes which forbid specified
                  difference sequences",
 booktitle     = "Global Telecommunications Conference, 1999. {GLOBECOM}~'99",
 address       = "Rio de Janeireo",
 year          = "1999",
 volume        = "1b",
 pages         = "878--882",
 doi           = "10.1109/GLOCOM.1999.830204",
 url           = "http://cmrr-star.ucsd.edu/static/pubs/globecom99_bruce.pdf",
 annote        = "Certain magnetic recording applications call for a large
                  number of sequences whose differences do not include
                  certain disallowed patterns. We show that the number of
                  such sequences increases exponentially with their length
                  and that the exponent, or capacity, is the logarithm of the
                  joint spectral radius of an appropriately defined set of
                  matrices. We derive new algorithms for determining the
                  joint spectral radius of sets of nonnegative matrices and
                  combine them with existing algorithms to determine the
                  capacity of several sets of disallowed differences that
                  arise in practice.",
}

@ARTICLE{MOS:IEEETIT01,
 author        = "Moision, Bruce E. and Orlitsky, Alon and Siegel, Paul H.",
 authauthor    = "Bruce Moision and Alon Orlitsky and Paul Siegel",
 title         = "On codes that avoid specified differences",
 journal       = "IEEE Trans. Inform. Theory",
 fjournal      = "Institute of Electrical and Electronics Engineers.
                  Transactions on Information Theory",
 year          = "2001",
 volume        = "47",
 number        = "1",
 pages         = "433--442",
 coden         = "IETTAW",
 issn          = "0018-9448",
 mrclass       = "94B65",
 mrnumber      = "1820392 (2001k:94084)",
 mrreviewer    = "Antoine Lobstein",
 zblnumber     = "0998.94563",
 doi           = "10.1109/18.904557",
 url           = "https://ieeexplore.ieee.org/document/904557",
 annote        = "Certain magnetic recording applications call for a large
                  number of sequences whose differences do not include
                  certain disallowed binary patterns. We show that the number
                  of such sequences increases exponentially with their length
                  and that the growth rate, or capacity, is the logarithm of
                  the joint spectral radius of an appropriately defined set
                  of matrices. We derive a new algorithm for determining the
                  joint spectral radius of sets of nonnegative matrices and
                  combine it with existing algorithms to determine the
                  capacity of several sets of disallowed differences that
                  arise in practice.",
}

@ARTICLE{MOS:LAA07,
 author        = "Moision, Bruce and Orlitsky, Alon and Siegel, Paul H.",
 authauthor    = "Moision, Bruce and Orlitsky, Alon and Siegel, Paul",
 title         = "On codes with local joint constraints",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2007",
 volume        = "422",
 number        = "2-3",
 pages         = "442--454",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "68R15 (15A60)",
 mrnumber      = "2305130 (2008f:68121)",
 mrreviewer    = "Sing Cheong Ong",
 zblnumber     = "1180.68207",
 doi           = "10.1016/j.laa.2006.11.002",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379506005039",
 annote        = "We study the largest number of sequences with the property
                  that any two sequences do not contain specified pairs of
                  patterns. We show that this number increases exponentially
                  with the length of the sequences and that the exponent, or
                  capacity, is the logarithm of the joint spectral radius of
                  an appropriately defined set of matrices. We illustrate a
                  new heuristic for computing the joint spectral radius and
                  use it to compute the capacity for several simple
                  collections. The problem of computing the achievable rate
                  region of a collection of codes is introduced and it is
                  shown that the region may be computed via a similar
                  analysis.",
}

@ARTICLE{Mojskerc:LAA14,
 author        = "Moj{\v{s}}kerc, Bla{\v{z}}",
 title         = "On the structure of finite-dimensional paracontractions",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2014",
 volume        = "446",
 pages         = "148--162",
 issn          = "0024-3795",
 mrclass       = "15A60",
 mrnumber      = "3163134",
 zblnumber     = "06277974",
 doi           = "10.1016/j.laa.2014.01.006",
 url           = "https://www.sciencedirect.com/science/article/pii/S002437951400024X",
 annote        = "A paracontraction with respect to a vector norm $\|\cdot\|$
                  is a matrix $A\in\mathcal{M}_{n}$ with the following
                  property: either $Ax=x$ or $\|Ax\|<\|x\|$ for any
                  $x\in\mathbb{C}^{n}$. Paracontractions arise naturally when
                  observing various stability properties of products of
                  matrices. \par We give a characterization of
                  paracontractions with respect to a strictly convex norm on
                  $\mathbb{C}^{n}$. As an application we give a
                  characterization of linear mappings on $\mathcal{M}_{n}$
                  which preserve paracontractions with respect to $p$-norms
                  for $1\le p\le\infty$ in both directions.",
}

@ARTICLE{MonMarg:AJIFAC11,
 author        = "Monovich, Tal and Margaliot, Michael",
 title         = "A second-order maximum principle for discrete-time bilinear
                  control systems with applications to discrete-time linear
                  switched systems",
 journal       = "Automatica J. IFAC",
 fjournal      = "Automatica. A Journal of IFAC, the International Federation
                  of Automatic Control",
 year          = "2011",
 volume        = "47",
 number        = "7",
 pages         = "1489--1495",
 coden         = "ATCAA9",
 issn          = "0005-1098",
 mrclass       = "49K21 (93C55)",
 mrnumber      = "2889248 (2012m:49046)",
 zblnumber     = "1220.49013",
 doi           = "10.1016/j.automatica.2011.02.025",
 url           = "https://www.sciencedirect.com/science/article/pii/S0005109811001269",
 annote        = "A powerful approach for analyzing the stability of
                  continuous-time switched systems is based on using optimal
                  control theory to characterize the ``most unstable''
                  switching law. This reduces the problem of determining
                  stability under arbitrary switching to analyzing stability
                  for the specific ``most unstable'' switching law. For
                  discrete-time switched systems, the variational approach
                  received considerably less attention. This approach is
                  based on using a first-order necessary optimality condition
                  in the form of a Maximum Principle (MP), and typically this
                  is not enough to completely characterize the ``most
                  unstable'' switching law. In this paper, we provide a
                  simple and self-contained derivation of a second-order
                  necessary optimality condition for discrete-time bilinear
                  control systems. This provides new information that cannot
                  be derived using the first-order MP. We demonstrate several
                  applications of this second-order MP to the stability
                  analysis of discrete-time linear switched systems.",
}

@ARTICLE{Morris:LAA10,
 author        = "Morris, Ian D.",
 authauthor    = "Ian Morris",
 title         = "Criteria for the stability of the finiteness property and
                  for the uniqueness of {B}arabanov norms",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2010",
 volume        = "433",
 number        = "7",
 pages         = "1301--1311",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15A60 (15A18)",
 mrnumber      = "2680257",
 zblnumber     = "1202.15028",
 zblreviewer   = "Alexey Alimov (Moskva)",
 doi           = "10.1016/j.laa.2010.05.007",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379510002545",
 annote        = "A set of matrices is said to have the \emph{finiteness
                  property} if the maximal rate of exponential growth of long
                  products of matrices drawn from that set is realized by a
                  periodic product. The extent to which the finiteness
                  property is prevalent among finite sets of matrices is the
                  subject of ongoing research. In this article, we give a
                  condition on a finite irreducible set of matrices which
                  guarantees that the finiteness property holds not only for
                  that set, but also for all sufficiently nearby sets of
                  equal cardinality. We also prove a theorem giving
                  conditions under which the Barabanov norm associated to a
                  finite irreducible set of matrices is unique up to
                  multiplication by a scalar, and show that in certain cases
                  these conditions are also persistent under small
                  perturbations.",
}

@ARTICLE{Morris:ADVM10,
 author        = "Morris, Ian D.",
 authauthor    = "Ian Morris",
 title         = "A rapidly-converging lower bound for the joint spectral
                  radius via multiplicative ergodic theory",
 journal       = "Adv. Math.",
 fjournal      = "Advances in Mathematics",
 year          = "2010",
 volume        = "225",
 number        = "6",
 pages         = "3425--3445",
 eprinttype    = "arXiv",
 eprint        = "0906.0260",
 coden         = "ADMTA4",
 issn          = "0001-8708",
 mrclass       = "37Hxx (37Axx 47A35)",
 mrnumber      = "2729011",
 zblnumber     = "1205.15032",
 zblreviewer   = "Tin Yau Tam (Auburn)",
 doi           = "10.1016/j.aim.2010.06.008",
 url           = "https://www.sciencedirect.com/science/article/pii/S0001870810002252",
 annote        = "We use ergodic theory to prove a quantitative version of a
                  theorem of \emph{M.\,A.~Berger} and
                  \emph{Y.~Wang}~\cite{BerWang:LAA92}, which relates the
                  joint spectral radius of a set of matrices to the spectral
                  radii of finite products of those matrices. The proof rests
                  on a theorem asserting the existence of a continuous
                  invariant splitting for certain matrix cocycles defined
                  over a minimal homeomorphism and having the property that
                  all forward products are uniformly bounded.\par Given
                  $\mathbf{A}$ a bounded nonempty set of $d\times d$ complex
                  matrices. The joint spectral radius of $\mathbf{A}$
                  introduced by Rota and Strang is \[\varrho(\mathbf{A}) :=
                  \lim_{n\to\infty}\sup\{\|A_n,\ldots A_1\|^{1/n}:
                  A_i\in\mathbf{A}\}\] where $\|\cdot\|$ denotes any norm on
                  $\mathbb{C}^d$. Berger--Wang formula asserts that
                  \[\varrho(\mathbf{A}) =
                  \lim_{n\to\infty}\sup\{\rho(A_n,\ldots A_1)^{1/n}:
                  A_i\in\mathbf{A}\}\] where $\rho(A)$ denotes the spectral
                  radius of a matrix $A$. The rate of convergence of
                  Berger--Wang formula is studied. A main result asserts that
                  for any positive real number $r$
                  \[\varrho(\mathbf{A})-\max_{1\le k\le n}
                  \varrho_k^-(\mathbf{A})=O\left(\frac 1{n^r}\right)\] where
                  \[\varrho_n^-(\mathbf{A}) := \sup\{\rho(A_n\cdots
                  A_1)^{1/n}: A_i\in\mathbf{A}\}.\] A more general result is
                  obtained when $\mathbf{A}$ is nonempty compact. The proof
                  rests on a structure theorem for continuous matrix cocycles
                  over minimal homomorphisms having the property that all
                  forward products are uniformly bounded. Possible extensions
                  are discussed. A comprehensive list of references is
                  given.",
}

@MISC{Morris:ArXiv11,
 author        = "Morris, Ian D.",
 authauthor    = "Morris, Ian",
 title         = "Rank one matrices do not contribute to the failure of the
                  finiteness property",
 howpublished  = "ArXiv.org e-Print archive",
 year          = "2011",
 month         = sep,
 eprinttype    = "arXiv",
 eprint        = "1109.4648",
 doi           = "10.48550/arXiv.1109.4648",
 annote        = "The joint spectral radius of a bounded set of $d\times d$
                  real or complex matrices is defined to be the maximum
                  exponential rate of growth of products of matrices drawn
                  from that set. A set of matrices is said to satisfy the
                  finiteness property if this maximum rate of growth occurs
                  along a periodic infinite sequence. In this note we give
                  some sufficient conditions for a finite set of matrices to
                  satisfy the finiteness property in terms of its rank one
                  elements. We show in particular that if a finite set of
                  matrices does not satisfy the finiteness property, then the
                  subset consisting of all matrices of rank at least two is
                  nonempty, does not satisfy the finiteness property, and has
                  the same joint spectral radius as the original set. We also
                  obtain an exact formula for the joint spectral radii of
                  sets of matrices which contain at most one element not of
                  rank one, generalising a recent result of X.~Dai.",
}

@ARTICLE{Morris:SIAMJMAA12,
 author        = "Morris, Ian D.",
 authauthor    = "Ian Morris",
 title         = "A new sufficient condition for the uniqueness of {B}arabanov
                  norms",
 journal       = "SIAM J. Matrix Anal. Appl.",
 fjournal      = "SIAM Journal on Matrix Analysis and Applications",
 year          = "2012",
 volume        = "33",
 number        = "2",
 pages         = "317--324",
 eprinttype    = "arXiv",
 eprint        = "1109.4649",
 coden         = "SJMAEL",
 issn          = "0895-4798",
 mrclass       = "15A60 (15A18)",
 mrnumber      = "2970208",
 mrreviewer    = "Luis Gonz{\'a}lez S{\'a}nchez",
 zblnumber     = "1256.15009",
 zblreviewer   = "Pei Yuan Wu (Hsinchu)",
 doi           = "10.1137/110837826",
 url           = "https://epubs.siam.org/doi/10.1137/110837826",
 annote        = "The joint spectral radius $\rho(\mathcal{A})$ of a bounded,
                  set, $\mathcal{A}$ of $d$-by-$d$ matrices over $\mathbb{K}$
                  ($=\mathbb{R}$ or $\mathbb{C}$) is defined by
                  \emph{G.-C.~Rota} and \emph{G.~Strang}~\cite{RotaStr:IM60},
                  to be $\limsup_{n\to\infty}\{\|A_{i_1}\cdots
                  A_{i_n}\|^{1/n}: A_{i_j}\in\mathcal{A}\}$. $\mathcal{A}$ is
                  said to be irreducible if there is no proper subspace of
                  $\mathbb{K}^d$ which is left invariant under all elements
                  of $\mathcal{A}$. \emph{N.\,E.~Barabanov}
                  showed~\cite{Bar:AIT88-2:e,Bar:AIT88-3:e,Bar:AIT88-5:e}
                  that, for any compact irreducible set $\mathcal{A}$ of
                  $M_d(\mathbb{K})$, there is associated a norm $|||.|||$ on
                  $\mathbb{K}^d$ such that $\rho(\mathcal{A})|||v|||=
                  \sup\{|||Av|||: A\in\mathcal{A}\}$ holds for all vectors
                  $v$ in $\mathbb{K}^d$. The present paper is concerned with
                  a sufficient condition for the uniqueness of such a
                  Barabanov norm for $\mathcal{A}$. The main result is as
                  follows:\par \textbf{Theorem 2.1.} \emph{If $\mathcal{A}$
                  is a nonempty bounded irreducible subset of
                  $M_d(\mathbb{K})$ such that the limit semigroup
                  $\&(\mathcal{A})\equiv\bigcap^\infty_{m=1}\,
                  (\overline{\bigcup^\infty_{n=m}
                  \rho(A)^{-n}\mathcal{A}^n})$, where $\mathcal{A}^n=
                  \{A_{i_1}\cdots A_{i_n}: A_{i_j}\in{\mathcal{A}}\}$, is
                  such that for every pair of nonzero vectors $v_1$ and $v_2$
                  in $\mathbb{K}^d$, there exist $B_1$ and $B_2$ in
                  $\&(\mathcal{A})$ and $\lambda$ in $\mathbb{K}$ such that
                  $B_1v_1=\lambda v_2$ and $B_2v_2= \lambda^{-1} v_1$, then
                  the Barabanov norm for $\mathcal{A}$ is unique up to a
                  positive scalar constant.}\par Examples of $\mathcal{A}$
                  consisting of 2-by-2 real matrices are given to illustrate
                  this condition and to contrast with the ones previously
                  obtained by the author. A theoretical application of the
                  main theorem is also given:\par \textbf{Theorem 4.1.}
                  \emph{There exists a nonempty open set $\mathcal{V} \subset
                  \mathcal{O}_2(\mathbb{R}^2)$ such that the sets
                  \[\mathcal{V}_1:=\left\{(A_1,A_2) \in\mathcal{V}\colon
                  \{A_1,A_2\}\text{ has a unique Barabanov norm}\right\},\]
                  \[\mathcal{V}_2:=\left\{(A_1,A_2) \in\mathcal{V}\colon
                  \{A_1,A_2\}\text{ does not have a unique Barabanov
                  norm}\right\}\] are both dense in $\mathcal{V}$.}\par This
                  is in contrast to a previous result of the author that
                  there is a nonempty open set $u$ of pairs of 2-by-2 real
                  matrices for which every pair in $u$ has a unique Barabanov
                  norm.",
}

@ARTICLE{Morris:JFA12,
 author        = "Morris, Ian D.",
 authauthor    = "Ian Morris",
 title         = "The generalised {B}erger-{W}ang formula and the spectral
                  radius of linear cocycles",
 journal       = "J. Funct. Anal.",
 fjournal      = "Journal of Functional Analysis",
 year          = "2012",
 volume        = "262",
 number        = "3",
 pages         = "811--824",
 eprinttype    = "arXiv",
 eprint        = "0906.2915",
 coden         = "JFUAAW",
 issn          = "0022-1236",
 mrclass       = "47A10",
 mrnumber      = "2863849",
 mrreviewer    = "Makoto Mori",
 zblnumber     = "1254.47006",
 zblreviewer   = "Andrzej So{\l}tysiak (Pozna{\'n})",
 doi           = "10.1016/j.jfa.2011.09.021",
 url           = "https://www.sciencedirect.com/science/article/pii/S002212361100365X",
 annote        = "Using multiplicative ergodic theory we prove two formulae
                  describing the relationships between different joint
                  spectral radii for sets of bounded linear operators acting
                  on a Banach space. In particular we recover a formula
                  previously proved by V.\,S.~Shulman and
                  Yu.\,V.~Turovski{\u\i} using operator-theoretic ideas. As a
                  byproduct of our method we answer a question of
                  J.\,E.~Cohen on the limiting behaviour of the spectral
                  radius of a measurable matrix cocycle.\par Let
                  $\mathcal{X}$ be a Banach space and let $B_\mathcal{X}$ be
                  its unit ball. The Hausdorff measure of noncompactness for
                  an operator $A\in \mathcal{B}(\mathcal{X})$ is defined to
                  be $\|A\|_\chi =\inf\{\varepsilon>0 : A(B_\mathcal{X})~
                  \text{is covered by a finite set of}
                  ~\varepsilon\text{-balls}\}$. The finite rank approximation
                  seminorm for $A$ is defined to be $\|A\|_f =\inf\{\|A-F\| :
                  \text{rank}\, F<\infty\}$. For a bounded set of operators
                  $\mathbf{A}$ on $\mathcal{X}$ let $\mathbf{A}^n
                  =\{A_{i_1}\cdots A_{i_n} : A_i \in \mathbf{A}\}$, $n\in
                  \mathbb{N}$. The authors consider the following joint
                  spectral radii: the Rota--Strang joint spectral radius
                  $\hat{\varrho}(\mathbf{A})=\lim_{n\to\infty}
                  \sup\{\|A\|^{1/n} : A\in \mathbf{A}^n \}$,
                  $\varrho_{\chi}(\mathbf{A})=\lim_{n\to\infty}
                  \sup\{\|A\|_{\chi}^{1/n} : A\in \mathbf{A}^n \}$,
                  $\varrho_{f}(\mathbf{A})=\lim_{n\to\infty}
                  \sup\{\|A\|_{f}^{1/n} : A\in \mathbf{A}^n\}$, and
                  $\varrho_{r}(\mathbf{A})=\limsup_{n\to\infty}
                  \sup\{\rho(A)^{1/n} : A\in \mathbf{A}^n \}$, where
                  $\rho(A)$ is the spectral radius of the operator $A$.\par
                  Let $T$ be a continuous transformation of a compact metric
                  space $X$ and $\mathcal{X}$ be a Banach space. A linear
                  cocycle over $T$ is a function $\mathcal{A} : X\times
                  \mathbb{N}\to\mathcal{B}(\mathcal{X})$ such
                  $\mathcal{A}(x,n+m)=\mathcal{A}(T^m x,n)\mathcal{A}(x,m)$
                  for all $x\in X$ and $n,m\in \mathbb{N}$. The main result
                  of the paper is the following:\par \textbf{Theorem.}
                  \emph{For a continuous cocycle $\mathcal{A}$ over $T$, the
                  following formulas hold true:} \[\lim_{n\to\infty}
                  \sup_{x\in X} \|\mathcal{A}(x,n)\|^{1/n}
                  =\max\left\{\limsup_{n\to\infty} \sup_{x\in X}
                  \rho(\mathcal{A}(x,n))^{1/n} ,~ \lim_{n\to\infty}
                  \sup_{x\in X} \|\mathcal{A}(x,n)\|_{\chi}^{1/n}\right\},\]
                  \[\lim_{n\to\infty} \sup_{x\in X}
                  \|\mathcal{A}(x,n)\|_{\chi}^{1/n} = \lim_{n\to\infty}
                  \sup_{x\in X} \|\mathcal{A}(x,n)\|_{f}^{1/n}.\] The
                  following formulas for the spectral radii of a precompact
                  and nonempty set $\mathbf{A}$ of operators on a Banach
                  space $\mathcal{X}$ are obtained as a consequence of this
                  theorem: \[\hat{\varrho}(\mathbf{A})=
                  \max\left\{\varrho_{\chi}(\mathbf{A}),~
                  \varrho_{r}(\mathbf{A})\right\},\quad
                  \varrho_{\chi}(\mathbf{A})=\varrho(\mathbf{A}).\] The first
                  of these formulas was proved earlier by different methods
                  by \emph{V.\,S.~Shulman} and
                  \emph{Y.\,V.~Turovskii}~\cite{ShulTur:JFA00,ShulTur:SM02}.",
}

@ARTICLE{Morris:PLMS13,
 author        = "Morris, Ian D.",
 authauthor    = "Morris, Ian",
 title         = "Mather sets for sequences of matrices and applications to
                  the study of joint spectral radii",
 journal       = "Proc. Lond. Math. Soc. (3)",
 fjournal      = "Proceedings of the London Mathematical Society. Third
                  Series",
 publisher     = "London Mathematical Society, London; Oxford University
                  Press, Oxford",
 year          = "2013",
 volume        = "107",
 number        = "1",
 pages         = "121--150",
 issn          = "0024-6115; 1460-244X/e",
 mrclass       = "37H15 (15A60)",
 mrnumber      = "3083190",
 mrreviewer    = "Lin Shu",
 zblnumber     = "06194569",
 doi           = "10.1112/plms/pds080",
 url           = "https://academic.oup.com/plms/article-abstract/107/1/121/1582314",
 annote        = "The joint spectral radius of a compact set of $d \times d$,
                  matrices is defined to be the maximum possible exponential
                  growth rate of products of matrices drawn from that set. In
                  this article we investigate the ergodic-theoretic structure
                  of those sequences of matrices drawn from a given set whose
                  products grow at the maximum possible rate. This leads to a
                  notion of Mather set for matrix sequences which is
                  analogous to the Mather set in Lagrangian dynamics. We
                  prove a structure theorem establishing the general
                  properties of these Mather sets and describing the extent
                  to which they characterise matrix sequences of maximum
                  growth. We give applications of this theorem to the study
                  of joint spectral radii and to the stability theory of
                  discrete linear inclusions.\par These results rest on some
                  general theorems on the structure of orbits of maximum
                  growth for subadditive observations of dynamical systems,
                  including an extension of the semi-uniform subadditive
                  ergodic theorem of Schreiber, Sturman and Stark, and an
                  extension of a noted lemma of Y.~Peres. These theorems are
                  presented in the appendix.",
}

@ARTICLE{Morris:AM16,
 author        = "Morris, Ian D.",
 authauthor    = "Morris, Ian",
 title         = "An inequality for the matrix pressure function and
                  applications",
 journal       = "Adv. Math.",
 fjournal      = "Advances in Mathematics",
 year          = "2016",
 volume        = "302",
 pages         = "280--308",
 eprinttype    = "arXiv",
 eprint        = "1507.00642",
 issn          = "0001-8708",
 mrclass       = "28A78 (15A30 15A60 82B30)",
 mrnumber      = "3545931",
 mrreviewer    = "Anouar Ben Mabrouk",
 zblnumber     = "1350.15005",
 doi           = "10.1016/j.aim.2016.07.025",
 url           = "https://www.sciencedirect.com/science/article/pii/S000187081530462X",
 annote        = "We prove an a priori lower bound for the pressure, or
                  $p$-norm joint spectral radius, of a measure on the set of
                  $d \times d$ real matrices which parallels a result of
                  \emph{J.~Bochi}~\cite{Bochi:LAA03} for the joint spectral
                  radius. We apply this lower bound to give new proofs of the
                  continuity of the affinity dimension of a self-affine set
                  and of the continuity of the singular-value pressure for
                  invertible matrices, both of which had been previously
                  established by D.-J.~Feng and P.~Shmerkin using
                  multiplicative ergodic theory and the subadditive
                  variational principle. Unlike the previous proof, our lower
                  bound yields algorithms to rigorously compute the pressure,
                  singular value pressure and affinity dimension of a finite
                  set of matrices to within an a priori prescribed accuracy
                  in finitely many computational steps. We additionally
                  deduce a related inequality for the singular value pressure
                  for measures on the set of $2 \times 2$ real matrices, give
                  a precise characterisation of the discontinuities of the
                  singular value pressure function for two-dimensional
                  matrices, and prove a general theorem relating the
                  zero-temperature limit of the matrix pressure to the joint
                  spectral radius.",
}

@ARTICLE{Morris:LAA17,
 author        = "Morris, Ian D.",
 authauthor    = "Ian Morris",
 title         = "Generic properties of the lower spectral radius for some
                  low-rank pairs of matrices",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2017",
 volume        = "524",
 pages         = "35--60",
 eprinttype    = "arXiv",
 eprint        = "1510.00209",
 issn          = "0024-3795",
 mrclass       = "15A60 (15A18 47D03)",
 mrnumber      = "3630178",
 zblnumber     = "1360.15025",
 doi           = "10.1016/j.laa.2017.02.023",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379517301222",
 annote        = "The lower spectral radius of a set of $d\times d$ matrices
                  is defined to be the minimum possible exponential growth
                  rate of long products of matrices drawn from that set. When
                  considered as a function of a finite set of matrices of
                  fixed cardinality it is known that the lower spectral
                  radius can vary discontinuously as a function of the matrix
                  entries. In a previous article the author and J. Bochi
                  conjectured that when considered as a function on the set
                  of all pairs of $2\times 2$ real matrices, the lower
                  spectral radius is discontinuous on a set of positive
                  (eight-dimensional) Lebesgue measure, and related this
                  result to an earlier conjecture of Bochi and Fayad. In this
                  article we investigate the continuity of the lower spectral
                  radius in a simplified context in which one of the two
                  matrices is assumed to be of rank one. We show in
                  particular that the set of discontinuities of the lower
                  spectral radius on the set of pairs of $2\times 2$ real
                  matrices has positive seven-dimensional Lebesgue measure,
                  and that among the pairs of matrices studied, the
                  finiteness property for the lower spectral radius is true
                  on a set of full Lebesgue measure but false on a residual
                  set.",
}

@ARTICLE{Morris:EDTS17-2,
 author        = "Morris, Ian D.",
 authauthor    = "Morris, Ian",
 title         = "Ergodic properties of matrix equilibrium states",
 journal       = "Ergodic Theory Dynam. Systems",
 fjournal      = "Ergodic Theory and Dynamical Systems",
 publisher     = "Cambridge University Press",
 year          = "2018",
 volume        = "38",
 number        = "6",
 pages         = "2295--2320",
 eprinttype    = "arXiv",
 eprint        = "1603.01744",
 issn          = "0143-3857",
 mrclass       = "37D35 (37H15)",
 mrnumber      = "3833350",
 mrreviewer    = "Xinsheng Wang",
 zblnumber     = "1397.37033",
 doi           = "10.1017/etds.2016.117",
 url           = "https://dx.doi.org/10.1017/etds.2016.117",
 annote        = "Given a finite irreducible set of real $d \times d$ matrices
                  $A_1,\ldots,A_M$ and a real parameter $s>0$, there exists a
                  unique shift-invariant equilibrium state on
                  $\{1,\ldots,M\}^{\mathbb{N}}$ associated to
                  $(A_1,\ldots,A_M,s)$. In this article we characterise the
                  ergodic properties of such equilibrium states in terms of
                  the algebraic properties of the semigroup generated by the
                  associated matrices. We completely characterise when the
                  equilibrium state has zero entropy, when it is fully
                  supported, when it gives distinct Lyapunov exponents to the
                  natural cocycle generated by $A_1,\ldots,A_M$, and when it
                  is a Bernoulli measure. We also give a general sufficient
                  condition for the equilibrium state to be mixing, and give
                  an example where the equilibrium state is ergodic but not
                  totally ergodic.",
}

@ARTICLE{Morris:EDTS17,
 author        = "Morris, Ian D.",
 authauthor    = "Morris, Ian",
 title         = "{L}yapunov-maximising measures for pairs of weighted shift
                  operators",
 journal       = "Ergodic Theory Dynam. Systems",
 fjournal      = "Ergodic Theory and Dynamical Systems",
 publisher     = "Cambridge University Press",
 year          = "2019",
 volume        = "39",
 number        = "1",
 pages         = "225--247",
 month         = jan,
 eprinttype    = "arXiv",
 eprint        = "1510.00162",
 issn          = "0143-3857",
 mrclass       = "37D25 (37A05 37A35 37H15 47A10)",
 mrnumber      = "3881131",
 mrreviewer    = "Miaohua Jiang",
 zblnumber     = "1402.37039",
 doi           = "10.1017/etds.2017.22",
 url           = "https://dx.doi.org/10.1017/etds.2017.22",
 annote        = "Motivated by recent investigations of ergodic optimisation
                  for matrix cocycles, we study the measures of maximum top
                  Lyapunov exponent for pairs of bounded weighted shift
                  operators on a separable Hilbert space. We prove that for
                  generic pairs of weighted shift operators the
                  Lyapunov-maximising measure is unique, and show that there
                  exist pairs of operators whose unique Lyapunov-maximising
                  measure takes any prescribed value less than $\log 2$ for
                  its metric entropy. We also show that in contrast to the
                  matrix case, the Lyapunov-maximising measures of pairs of
                  bounded operators are in general not characterised by their
                  supports: we construct explicitly a pair of operators, and
                  a pair of ergodic measures on the 2-shift with identical
                  supports, such that one of the two measures is
                  Lyapunov-maximising for the pair of operators and the other
                  measure is not. Our proofs make use of the Ornstein
                  $\overline{d}$-metric to estimate differences in the top
                  Lyapunov exponent of a pair of weighted shift operators as
                  the underlying measure is varied.",
}

@ARTICLE{Morris:SIAMJMAA22,
 author        = "Morris, Ian D.",
 authauthor    = "Morris, Ian",
 title         = "Fast approximation of the {$p$}-radius, matrix pressure, or
                  generalized {L}yapunov exponent for positive and dominated
                  matrices",
 journal       = "SIAM J. Matrix Anal. Appl.",
 fjournal      = "SIAM Journal on Matrix Analysis and Applications",
 year          = "2022",
 volume        = "43",
 number        = "1",
 pages         = "178--198",
 eprinttype    = "arXiv",
 eprint        = "1905.00749",
 issn          = "0895-4798",
 mrclass       = "15A60 (37H15)",
 mrnumber      = "4378590",
 zblnumber     = "1487.37099",
 zblreviewer   = "Matheus Cheque Bortolan (Florian{\'o}polis)",
 doi           = "10.1137/19M1303964",
 url           = "https://epubs.siam.org/doi/10.1137/19M1303964",
 annote        = "If $A_1,\ldots,A_N$ are real $d \times d$ matrices, the
                  \emph{$p$-radius}, \emph{generalised Lyapunov exponent} or
                  \emph{matrix pressure} is defined to be the asymptotic
                  exponential growth rate of the sum
                  $\sum_{i_1,\ldots,i_n=1}^N \|A_{i_n}\cdots A_{i_1}\|^p$,
                  where $p$ is a real parameter. Under its various names this
                  quantity has been investigated for its applications to
                  topics including wavelet regularity and refinement
                  equations, fractal geometry and the large deviations theory
                  of random matrix products. In this article we present a new
                  algorithm for computing the $p$-radius under the hypothesis
                  that the matrices are all positive, or more generally under
                  the hypothesis that they satisfy a weaker condition called
                  \emph{domination}. This algorithm is based on interpreting
                  the $p$-radius as the leading eigenvalue of a trace-class
                  operator on a Hilbert space and estimating that eigenvalue
                  via approximations to the Fredholm determinant of the
                  operator. In this respect our method is closely related to
                  the work of Z.-Q. Bai and M. Pollicott on computing the top
                  Lyapunov exponent of a random matrix product. For pairs of
                  positive matrices of low dimension our method yields
                  substantial improvements over existing methods.",
}

@ARTICLE{Morris:SCL22,
 author        = "Morris, Ian D.",
 authauthor    = "Morris, Ian",
 title         = "Marginally unstable discrete-time linear switched systems
                  with highly irregular trajectory growth",
 journal       = "Systems Control Lett.",
 fjournal      = "Systems \&{} Control Letters",
 year          = "2022",
 volume        = "163",
 pages         = "Paper No. 105216, 8",
 issn          = "0167-6911",
 mrclass       = "39A99 (34A38 34D20)",
 mrnumber      = "4409298",
 zblnumber     = "07535816",
 eprinttype    = "arXiv",
 eprint        = "2111.10225",
 doi           = "10.1016/j.sysconle.2022.105216",
 url           = "https://www.sciencedirect.com/science/article/pii/S0167691122000512",
 annote        = "We investigate the uniform stability properties of
                  discrete-time linear switched systems subject to arbitrary
                  switching, focusing on the ``marginally unstable'' regime
                  in which the system is not Lyapunov stable but in which
                  trajectories cannot escape to infinity at exponential
                  speed. For a discrete linear system of this type without
                  switching the fastest-growing trajectory must grow as an
                  exact polynomial function of time, and a significant body
                  of prior research has focused on investigating how far this
                  intuitive picture can be extended from systems without
                  switching to cases where switching is present. In this note
                  we give an example of a family of discrete linear switched
                  systems in three dimensions, with two switching states, for
                  which this intuition fails badly: for a generic member of
                  this family the maximal rate of uniform growth of escaping
                  trajectories can be made arbitrarily slow along one
                  subsequence of times and yet also faster than any
                  prescribed slower-than-linear function along a
                  complementary subsequence of times. Using this construction
                  we give new counterexamples to a conjecture of
                  \emph{Y.~Chitour, P.~Mason,} and
                  \emph{M.~Sigalotti}~\cite{CMS:SCL12} and obtain a negative
                  answer to a related question of \emph{R.~Jungers,
                  V.~Protasov} and \emph{V.~Blondel}~\cite{JPB:LAA08}. Our
                  examples have the additional feature that marginal
                  stability and marginal instability are densely intermingled
                  in the same parameter space.",
}

@ARTICLE{Morris:SIAMJCO24-2,
 author        = "Morris, Ian D.",
 authauthor    = "Morris, Ian",
 title         = "A Stability Dichotomy for Discrete-Time Linear Switching
                  Systems in Dimension Two",
 journal       = "SIAM J. Control Optim.",
 fjournal      = "SIAM Journal on Control and Optimization",
 volume        = "62",
 year          = "2024",
 number        = "1",
 pages         = "400--414",
 issn          = "0363-0129,1095-7138",
 mrclass       = "99-06",
 mrnumber      = "4704593",
 zblnumber     = "07800996",
 eprinttype    = "arXiv",
 eprint        = "2301.06821",
 doi           = "10.1137/23M1551225",
 url           = "https://epubs.siam.org/doi/10.1137/23M1551225",
 annote        = "We prove that for every discrete-time linear switching
                  system in two complex variables and with finitely many
                  switching states, either the system is Lyapunov stable or
                  there exists a trajectory which escapes to infinity with at
                  least linear speed. We also give a checkable algebraic
                  criterion to distinguish these two cases. This dichotomy
                  was previously known to hold for systems in two real
                  variables, but is known to be false in higher dimensions
                  and for systems with infinitely many switching states.",
}

@ARTICLE{Morris:SIAMJCO24,
 author        = "Morris, Ian D.",
 authauthor    = "Morris, Ian",
 title         = "An irreducible linear switching system whose unique
                  {B}arabanov norm is not strictly convex",
 journal       = "SIAM J. Control Optim.",
 fjournal      = "SIAM Journal on Control and Optimization",
 volume        = "62",
 year          = "2024",
 number        = "1",
 pages         = "42--64",
 mrclass       = "93 (34)",
 mrnumber      = "4686843",
 zblnumber     = "07791448",
 eprinttype    = "arXiv",
 eprint        = "2301.09942",
 doi           = "10.1137/23M1551213",
 url           = "https://epubs.siam.org/doi/10.1137/23M1551213",
 annote        = "We construct a marginally stable linear switching system in
                  continuous time, in four dimensions and with three
                  switching states, which is exponentially stable with
                  respect to constant switching laws and which has a unique
                  Barabanov norm, but such that the Barabanov norm fails to
                  be strictly convex. This resolves a question of \emph{Y.
                  Chitour, M. Gaye} and \emph{P. Mason}~\cite{CGM:JDE15}.",
}

@ARTICLE{MorVar:DS23,
 author        = "Morris, Ian D. and Varney, Jonah",
 authauthor    = "Morris, Ian and Varney, Jonah",
 title         = "A note on the marginal instability rates of two-dimensional
                  linear cocycles",
 journal       = "Dyn. Syst.",
 fjournal      = "Dynamical Systems. An International Journal",
 volume        = "38",
 year          = "2023",
 number        = "4",
 pages         = "525--540",
 issn          = "1468-9367,1468-9375",
 mrclass       = "37H15 (37D35 93C30)",
 mrnumber      = "4653858",
 zblnumber     = "07758044",
 eprinttype    = "arXiv",
 eprint        = "2301.05152",
 doi           = "10.1080/14689367.2023.2210518",
 url           = "https://www.tandfonline.com/doi/abs/10.1080/14689367.2023.2210518",
 annote        = "Define $\Sigma_N:=\{1,\ldots,N\}^{\mathbb{Z}}$ and equip
                  this set with the infinite product topology, with respect
                  to which it is a compact metrisable topological space.
                  Define $T \colon \Sigma_N \to \Sigma_N$ to be the shift
                  transformation $T[(x_n)_{n \in \mathbb{Z}}]:=(x_{n+1})_{n
                  \in \mathbb{Z}}$, which is a homeomorphism. If a continuous
                  function $A \colon \Sigma_N \to
                  \mathop{\mathrm{GL}}_d(\mathbb{R})$ is specified, one may
                  be interested in the growth of the sequence $(a_n)$ defined
                  by \[a_n:= \sup_{x \in \Sigma_N} \left\|A(T^{n-1}x)\cdots
                  A(T x)A(x)\right\|.\] We give a proof of the following
                  theorem and demonstrate that its conclusions do not hold
                  for Lipschitz continuous cocycles over the full shift on
                  two symbols:\par \textbf{Theorem.} \emph{Let $A\colon
                  \Sigma_N \to {\mathrm{GL}}_2(\mathbb{R})$ be locally
                  constant and define $\mathsf{A}:=\{A(x) \colon x \in
                  \Sigma_N\}$. Suppose that \[\lim_{n \to \infty} \sup_{x \in
                  \Sigma_N} \left\|A(T^{n-1}x)\cdots
                  A(x)\right\|^{\frac{1}{n}}=1.\] Then one of the following
                  holds: either \[\lim_{n \to \infty}\frac{1}{n} \sup_{x \in
                  \Sigma_N} \left\|A(T^{n-1}x)\cdots A(x)\right\|>0,\] or we
                  instead have \[\sup_{n \geq 1} \sup_{x \in \Sigma_N}
                  \left\|A(T^{n-1}x)\cdots A(x)\right\|<\infty.\] Moreover,
                  the first case occurs if and only if the semigroup
                  generated by $\mathsf{A}$ contains a nontrivial Jordan
                  matrix with unit determinant, if and only if both of the
                  following two conditions are met: $\mathsf{A}$ is
                  simultaneously triangularisable, and the set of matrices in
                  $\mathsf{A}$ with determinant $\pm 1$ is nonempty and is
                  not simultaneously diagonalisable.}\par This theorem
                  implies that if the uniform norm growth of a locally
                  constant ${\mathrm{GL}}_2(\mathbb{R})$-cocycle on the full
                  shift is not exponential then it must be either bounded or
                  linear, with no other possibilities occurring.",
}

@ARTICLE{MorSid:JEMS13,
 author        = "Morris, Ian and Sidorov, Nikita",
 authauthor    = "Ian Morris and Nikita Sidorov",
 title         = "On a {D}evil's staircase associated to the joint spectral
                  radii of a family of pairs of matrices",
 journal       = "J. Eur. Math. Soc. (JEMS)",
 fjournal      = "Journal of the European Mathematical Society (JEMS)",
 year          = "2013",
 volume        = "15",
 number        = "5",
 pages         = "1747--1782",
 eprinttype    = "arXiv",
 eprint        = "1107.3506",
 issn          = "1435-9855",
 mrclass       = "Preliminary Data",
 mrnumber      = "3082242",
 zblnumber     = "06203606",
 doi           = "10.4171/JEMS/402",
 url           = "https://www.ems-ph.org/journals/show_abstract.php?issn=1435-9855&vol=15&iss=5&rank=5",
 annote        = "The joint spectral radius of a finite set of real $d\times
                  d$ matrices is defined to be the maximum possible
                  exponential rate of growth of products of matrices drawn
                  from that set. In previous work with K.\,G.~Hare and
                  J.~Theys we showed that for a certain one-parameter family
                  of pairs of matrices, this maximum possible rate of growth
                  is attained along Sturmian sequences with a certain
                  characteristic ratio which depends continuously upon the
                  parameter. In this paper we answer some open questions from
                  that paper by showing that the dependence of the ratio
                  function upon the parameter takes the form of a Devil's
                  staircase. We show in particular that this Devil's
                  staircase attains every rational value strictly between 0
                  and 1 on some interval, and attains irrational values only
                  in a set of Hausdorff dimension zero. This result
                  generalises to include certain one-parameter families
                  considered by other authors. We also give explicit formulas
                  for the preimages of both rational and irrational numbers
                  under the ratio function, thereby establishing a large
                  family of pairs of matrices for which the joint spectral
                  radius may be calculated exactly.",
}

@ARTICLE{Muller:APM97,
 author        = "M{\"u}ller, Vladim{\'\i}r",
 title         = "On the joint spectral radius",
 journal       = "Ann. Polon. Math.",
 fjournal      = "Annales Polonici Mathematici",
 year          = "1997",
 volume        = "66",
 pages         = "173--182",
 coden         = "APNMA4",
 issn          = "0066-2216",
 mrclass       = "46H05 (46J05 47A13)",
 mrnumber      = "1438337 (98e:46058)",
 mrreviewer    = "Marc Thomas",
 zblnumber     = "0877.46037",
 url           = "https://users.math.cas.cz/~muller/papers.htm",
 note          = "Volume dedicated to the memory of W{\l}odzimierz Mlak",
 annote        = "We prove the $\ell_p$-spectral radius formula for $n$-tuples
                  of commuting Banach algebra elements. This generalizes
                  results of some earlier papers.",
}

@ARTICLE{MulPep:LAA13,
 author        = "M{\"u}ller, Vladimir and Peperko, Aljo{\v{s}}a",
 title         = "Generalized spectral radius and its max algebra version",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2013",
 volume        = "439",
 number        = "4",
 pages         = "1006--1016",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15A60 (15A18 15A80 15B48)",
 mrnumber      = "3061751",
 mrreviewer    = "Buket Benek Gursoy",
 zblnumber     = "1281.15009",
 doi           = "10.1016/j.laa.2012.09.024",
 url           = "https://www.sciencedirect.com/science/article/pii/S00243795120073804",
 annote        = "Let $\Sigma\subset\mathbb{C}^{n\times n}$ and $\Psi\subset
                  \mathbb{R}^{n\times n}_{+}$ be bounded subsets and let
                  $\rho(\Sigma)$ and $\mu(\Psi)$ denote the generalized
                  spectral radius of $\Sigma$ and the max algebra version of
                  the generalized spectral radius of $\Psi$, respectively. We
                  apply a single matrix description of $\mu(\Psi)$ to give a
                  new elementary and straightforward proof of the
                  Berger--Wang formula in max algebra and consequently a new
                  short proof of the original Berger--Wang formula in the
                  case of bounded subsets of $n\times n$ non-negative
                  matrices. We also obtain a new description of $\mu(\Psi)$
                  in terms of the Schur--Hadamard product and prove new trace
                  and max-trace descriptions of $\mu(\Psi)$ and
                  $\rho(\Sigma)$. In particular, we show that \[\mu(\Psi)=
                  \limsup_{m\to\infty}~[\sup_{A\in\Psi^{m}_{\otimes}}
                  \mathrm{tr}_{\otimes}(A)]^{1/m} =
                  \limsup_{m\to\infty}~[\sup_{A\in\Psi^{m}_{\otimes}}
                  \mathrm{tr}(A)]^{1/m}\] and \[\rho(\Sigma)=
                  \limsup_{m\to\infty}~[\sup_{B\in\Sigma^{m}}
                  \mathrm{tr}(|B|)]^{1/m}=
                  \limsup_{m\to\infty}~[\sup_{B\in\Sigma^{m}}
                  \mathrm{tr}_{\otimes}(|B|)]^{1/m}\] where
                  $\mathrm{tr}_{\otimes}(A) = \max_{i=1,\ldots,n} a_{ii}$ and
                  $|B| = [|b_{ij}|]$.",
}

@ARTICLE{MP:DCDCA18,
 author        = "M{\"u}ller, Vladimir and Peperko, Aljo{\v{s}}a",
 title         = "Lower spectral radius and spectral mapping theorem for
                  suprema preserving mappings",
 journal       = "Discrete Contin. Dyn. Syst.",
 fjournal      = "Discrete and Continuous Dynamical Systems. Series A",
 year          = "2018",
 volume        = "38",
 number        = "8",
 pages         = "4117--4132",
 eprinttype    = "arXiv",
 eprint        = "1712.00340",
 issn          = "1078-0947",
 mrclass       = "47H07 (47A10 47B65 47J10)",
 mrnumber      = "3814367",
 mrreviewer    = "J{\"o}rg Eschmeier",
 zblnumber     = "06919411",
 doi           = "10.3934/dcds.2018179",
 url           = "https://www.aimsciences.org/article/doi/10.3934/dcds.2018179",
 annote        = "We study Lipschitz, positively homogeneous and finite
                  suprema preserving mappings defined on a max-cone of
                  positive elements in a normed vector lattice. We prove that
                  the lower spectral radius of such a mapping is always a
                  minimum value of its approximate point spectrum. We apply
                  this result to show that the spectral mapping theorem holds
                  for the approximate point spectrum of such a mapping. By
                  applying this spectral mapping theorem we obtain new
                  inequalites for the Bonsall cone spectral radius of max
                  type kernel operators.",
}

@ARTICLE{Mus:SM23:e,
 author        = "Musaeva, Asiyat Magomedovna",
 title         = "Construction of invariant {L}yapunov norms of planar
                  dynamical systems",
 fjournal      = "Matematicheski{\u\i} Sbornik",
 journal       = "Sb. Math",
 year          = "2023",
 volume        = "214",
 number        = "9",
 pages         = "1212--1240",
 url           = "https://mi.mathnet.ru/eng/sm9821",
 doi           = "10.4213/sm9821e",
 mrclass       = "34D20 (34A30 34D08 37N35 39A22)",
 mrnumber      = "4705898",
 language      = "english",
 annote        = "We consider the problem of the stability of linear dynamical
                  switching systems. It is known that an irreducible
                  $d$-dimensional system always has an invariant Lyapunov
                  norm (a Barabanov norm), which determines the stability of
                  the system and the rate of growth of its trajectories. We
                  prove that in the case of $d=2$ the invariant norm is a
                  piecewise analytic function and can be constructed
                  explicitly for every finite system of matrices. The method
                  of construction, an algorithm for computing the Lyapunov
                  exponent and a way for deciding the stability of the system
                  are presented. A complete classification of invariant norms
                  for planar systems is derived. A criterion of the
                  uniqueness of an invariant norm of a given system is
                  proved, and some special norms (norms generated by polygons
                  and so on) are investigated.",
}

@ARTICLE{Mus:SM23:r,
 author        = "Musaeva, Asiyat Magomedovna",
 title         = "Construction of invariant {L}yapunov norms of planar
                  dynamical systems",
 journal       = "Sbornik: Mathematics",
 year          = "2023",
 volume        = "214",
 number        = "9",
 pages         = "27--57",
 url           = "http://mi.mathnet.ru/sm9821",
 doi           = "10.4213/sm9821",
 language      = "english",
 note          = "in Russian",
 annote        = "The problem of stability of linear dynamic systems with
                  switchings is considered. It is known that an irreducible
                  $d$-dimensional system always has an invariant Lyapunov norm
                  (Barabanov norm), which determines the stability of the
                  system and the order of growth of its trajectories. We
                  prove that in the case of $d=2$ the invariant norm is a
                  piecewise analytic function and can be constructed
                  explicitly for any system with a finite number of matrices.
                  A construction method, an algorithm for calculating the
                  Lyapunov exponent, and a method for determining the
                  stability of the system are presented. A complete
                  classification of invariant norms of planar systems is
                  obtained. A criterion for the uniqueness of an invariant
                  norm for a given system has been proven, and norms of a
                  special type have been studied (norms generated by
                  polygons, etc.).",
}

@ARTICLE{NesPro:SIAMJMAA13,
 author        = "Nesterov, {\relax{}Yu}. and Protasov, V. {\relax{}Yu}.",
 authauthor    = "Nesterov, {\relax{}Yu}. and Protasov, V.",
 title         = "Optimizing the spectral radius",
 journal       = "SIAM J. Matrix Anal. Appl.",
 fjournal      = "SIAM Journal on Matrix Analysis and Applications",
 year          = "2013",
 volume        = "34",
 number        = "3",
 pages         = "999--1013",
 coden         = "SJMAEL",
 issn          = "0895-4798",
 mrclass       = "Preliminary Data",
 mrnumber      = "3073651",
 zblnumber     = "1282.15029",
 doi           = "10.1137/110850967",
 url           = "https://epubs.siam.org/doi/10.1137/110850967",
 annote        = "We suggest a new approach to finding the maximal and the
                  minimal spectral radii of linear operators from a given
                  compact family of operators, which share a common invariant
                  cone (e.g., family of nonnegative matrices). In the case of
                  families with the so-called product structure, this leads
                  to efficient algorithms for optimizing the spectral radius
                  and for finding the joint and lower spectral radii of the
                  family. Applications to the theory of difference equations
                  and to problems of optimizing the spectral radius of graphs
                  are considered.",
}

@ARTICLE{NP:SIAMJMAA20,
 author        = "Nesterov, {\relax{}Yu}. and Protasov, V. {\relax{}Yu}.",
 authauthor    = "Nesterov, {\relax{}Yu}rii and Protasov, Vladimir",
 title         = "Computing closest stable nonnegative matrix",
 journal       = "SIAM J. Matrix Anal. Appl.",
 fjournal      = "SIAM Journal on Matrix Analysis and Applications",
 year          = "2020",
 volume        = "41",
 number        = "1",
 pages         = "1--28",
 issn          = "0895-4798",
 mrclass       = "65F15 (15B48 90C26 93D09)",
 mrnumber      = "4048011",
 zblnumber     = "1448.65027",
 doi           = "10.1137/17M1144568",
 url           = "https://epubs.siam.org/doi/abs/10.1137/17M1144568",
 annote        = "Problem of finding the closest stable matrix for a dynamical
                  system has many applications. It is well studied both for
                  continuous and discrete-time systems, and the corresponding
                  optimization problems are formulated for various matrix
                  norms. As a rule, non-convexity of these formulations does
                  not allow finding their global solutions. In this paper, we
                  analyze positive discrete-time systems. They also suffer
                  from non-convexity of the stability region, and the problem
                  in the Frobenius norm or in Euclidean norm remains hard for
                  them. However, it turns out that for certain polyhedral
                  norms, the situation is much better. We show, that for the
                  distances measured in the max-norm, we can find exact
                  solution of the corresponding nonconvex projection problems
                  in polynomial time. For the distance measured in the
                  operator $\ell_{\infty}$-norm or $\ell_{1}$-norm, the exact
                  solution is also efficiently found. To this end, we develop
                  a modification of the recently introduced spectral simplex
                  method. On the other hand, for all these three norms, we
                  obtain exact descriptions of the region of stability around
                  a given stable matrix. In the case of max-norm, this can be
                  seen as an analogue of Kharitonov's theorem for
                  non-negative matrices.",
}

@MISC{NeuSch:ArXiv98,
 author        = "Neumann, Michael and Schneider, Hans",
 title         = "Partial norms and the convergence of general products of
                  matrices",
 howpublished  = "ArXiv.org e-Print archive",
 year          = "1998",
 month         = feb,
 eprinttype    = "arXiv",
 eprint        = "math/9802108",
 doi           = "10.48550/arXiv.math/9802108",
 annote        = "Motivated by the theory of inhomogeneous Markov chains, we
                  determine a sufficient condition for the convergence to $0$
                  of a general product formed from a sequence of real or
                  complex matrices. When the matrices have a common invariant
                  subspace $H$, we give a sufficient condition for the
                  convergence to $0$ on $H$ of a general product. Our result
                  is applied to obtain a condition for the weak ergodicity of
                  an inhomogeneous Markov chain. We compare various types of
                  contractions which may be defined for a single matrix, such
                  as paracontraction, $l$--contraction, and $H$--contraction,
                  where $H$ is an invariant subspace of the matrix.",
}

@ARTICLE{NSch:LAA99,
 author        = "Neumann, Michael and Schneider, Hans",
 title         = "The convergence of general products of matrices and the weak
                  ergodicity of {M}arkov chains",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "1999",
 volume        = "287",
 number        = "1-3",
 pages         = "307--314",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15A60 (15A51 60J10)",
 mrnumber      = "1662874 (2000a:15047)",
 mrreviewer    = "D. J. Hartfiel",
 zblnumber     = "0943.15011",
 zblreviewer   = "Alexei Khorunzhy (Khar'kov)",
 doi           = "10.1016/S0024-3795(98)10196-9",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379598101969",
 note          = "Special issue celebrating the 60th birthday of Ludwig
                  Elsner",
 annote        = "The paper concerns the infinite general products of real or
                  complex matrices. General means that the matrices to form
                  the product are taken in arbitrary order from an infinite
                  sequence of matrices $\{A_i\}$. In the case of stochastic
                  matrices, such general products have been considered by
                  \emph{E.~Seneta}~\cite{Seneta06}. \par In two main
                  theorems, the authors give sufficient conditions for such a
                  product to be bounded and to be convergent to zero,
                  respectively. These conditions use the matrix
                  submultiplicative norm $\mu$ and are based on the
                  convergence of $\sum_i [\max(\mu(A_i),1)-1]$ and divergence
                  of $\sum_i [1-\min(\mu(A_i),1)]$, respectively. \par The
                  proofs are based on classical theorems related with the
                  products of positive numbers. The results obtained are used
                  to deduce a condition for weak ergodicity of inhomogeneous
                  Markov chain. Also the ergodicity coefficient based on the
                  matrix norm is compared with other ergodicity coefficients
                  considered, for example, in a paper by \emph{A. Rhodius}
                  [Linear Algebra Appl. 194, 71--83 (1993)].",
}

@ARTICLE{OgurMart:LAA13,
 author        = "Ogura, M. and Martin, C. F.",
 authauthor    = "Ogura, M. and Martin, C.",
 title         = "Generalized joint spectral radius and stability of switching
                  systems",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2013",
 volume        = "439",
 number        = "8",
 pages         = "2222--2239",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "Preliminary Data",
 mrnumber      = "3091300",
 zblnumber     = "1280.93090",
 doi           = "10.1016/j.laa.2013.06.028",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379513004278",
 annote        = "This paper extends the notion of generalized joint spectral
                  radius with exponents, originally defined for a finite set
                  of matrices, to probability distributions. We show that,
                  under a certain invariance condition, the radius is
                  calculated as the spectral radius of a matrix that can be
                  easily computed, extending the classical counterpart. Using
                  this result we investigate the mean stability of switching
                  systems. In particular we establish the equivalence of mean
                  square stability, simultaneous contractibility in square
                  mean, and the existence of a quadratic Lyapunov function.
                  Also the stabilization of positive switching systems is
                  studied. Numerical examples are given to illustrate the
                  results.",
}

@INPROCEEDINGS{OgurJung:CDC14,
 author        = "Ogura, Masaki and Jungers, Rapha{\"e} M.",
 authauthor    = "Ogura, Masaki and Jungers, Rapha{\"e}",
 title         = "Efficiently computable lower bounds for the {$p$}-radius of
                  switching linear systems",
 booktitle     = "53rd IEEE Conference on Decision and Control (CDC), 15-17
                  Dec.",
 publisher     = "IEEE",
 address       = "Los Angeles, California, USA",
 year          = "2014",
 pages         = "5463--5468",
 doi           = "10.1109/CDC.2014.7040243",
 url           = "https://ieeexplore.ieee.org/document/7040243/",
 annote        = "This paper proposes novel lower bounds on a quantity called
                  $L^{p}$-norm joint spectral radius, or in short,
                  $p$-radius, of a finite set of matrices. Despite its wide
                  range of applications, (for example, to the stability of
                  switching linear systems and the uniqueness of the
                  equilibrium solutions of switching linear economical
                  models), algorithms for computing the $p$-radius are only
                  available in a very limited number of particular cases. We
                  propose lower bounds that do not require any special
                  structure on matrices and are formulated as the maximal
                  spectral radius of a matrix family generated by weighting
                  matrices via Kronecker products. We show on numerical
                  examples that the proposed lower bounds can largely improve
                  the existing ones.",
}

@INPROCEEDINGS{OgurMart:CDC13,
 author        = "Ogura, Masaki and Martin, Clyde F.",
 authauthor    = "Ogura, Masaki and Martin, Clyde",
 title         = "On the Mean Stability of a Class of Switched Linear
                  Systems",
 booktitle     = "Proceedings of the 52th IEEE Conference on Decision and
                  Control, CDC~2013",
 year          = "2013",
 pages         = "97--102",
 eprinttype    = "arXiv",
 eprint        = "1409.6032",
 url           = "https://www.groundai.com/project/on-the-mean-stability-of-a-class-of-switched-linear-systems/1",
 annote        = "This paper investigates the mean stability of a class of
                  discrete-time stochastic switched linear systems using the
                  $L^p$-norm joint spectral radius of the probability
                  distributions governing the switched systems. First we
                  prove a converse Lyapunov theorem that shows the
                  equivalence between the mean stability and the existence of
                  a homogeneous Lyapunov function. Then we show that, when
                  $p$ goes to $\infty$, the stability of the $p$th mean
                  becomes equivalent to the absolute asymptotic stability of
                  an associated deterministic switched system. Finally we
                  study the mean stability of Markovian switched systems.
                  Numerical examples are presented to illustrate the
                  results.",
}

@INPROCEEDINGS{OgurMart:ECC13,
 author        = "Ogura, Masaki and Martin, Clyde F.",
 authauthor    = "Ogura, Masaki and Martin, Clyde",
 title         = "Stability of switching systems and generalized joint
                  spectral radius",
 booktitle     = "Proceedings of the European Control Conference, ECC~2013,
                  17--19 July",
 publisher     = "IEEE",
 address       = "Zurich, Switzerland",
 year          = "2013",
 pages         = "3185--3190",
 doi           = "10.23919/ECC.2013.6669115",
 url           = "https://ieeexplore.ieee.org/document/6669115",
 annote        = "This paper studies the mean stability of stochastic
                  switching linear systems. We first show that the mean
                  stability is characterized by an extended version of so
                  called generalized joint spectral radius. Then it is shown
                  that, under an invariance condition, the quantity can be
                  computed as the spectral radius of a certain matrix
                  associated with the given switching system. Also we show
                  that the mean square stability is equivalent to the
                  existence of a Lyapunov function. Our results are
                  illustrated by numerical examples.",
}

@ARTICLE{OgurMart:LAA14,
 author        = "Ogura, Masaki and Martin, Clyde F.",
 authauthor    = "Ogura, Masaki and Martin, Clyde",
 title         = "A limit formula for joint spectral radius with $p$-radius of
                  probability distributions",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2014",
 volume        = "458",
 pages         = "605--625",
 month         = oct,
 eprinttype    = "arXiv",
 eprint        = "1401.3026",
 issn          = "0024-3795",
 mrclass       = "15A60 (60B99 93E15)",
 mrnumber      = "3231838",
 mrreviewer    = "Janko Bra{\v{c}}i{\v{c}}",
 zblnumber     = "1294.15015",
 doi           = "10.1016/j.laa.2014.06.034",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379514004030",
 annote        = "In this paper we show a characterization of the joint
                  spectral radius of a set of matrices as the limit of the
                  $p$-radius of an associated probability distribution when
                  $p$ tends to $\infty$. Allowing the set to have infinitely
                  many matrices, the obtained formula extends the results in
                  the literature. Based on the formula, we then present a
                  novel characterization of the stability of switched linear
                  systems for an arbitrary switching signal via the existence
                  of stochastic Lyapunov functions of any higher degrees.
                  Numerical examples are presented to illustrate the
                  results.",
}

@ARTICLE{OPJ:SCL16,
 author        = "Ogura, Masaki and Preciado, Victor M. and Jungers,
                  Rapha{\"e}l M.",
 authauthor    = "Ogura, Masaki and Preciado, Victor and Jungers,
                  Rapha{\"e}l",
 title         = "Efficient method for computing lower bounds on the
                  {$p$}-radius of switched linear systems",
 journal       = "Systems Control Lett.",
 fjournal      = "Systems {\&} Control Letters",
 year          = "2016",
 volume        = "94",
 pages         = "159--164",
 eprinttype    = "arXiv",
 eprint        = "1503.03034",
 issn          = "0167-6911",
 mrclass       = "93E15 (15A60 34A38 60J10)",
 mrnumber      = "3530610",
 mrreviewer    = "Yonghui Sun",
 zblnumber     = "1344.93050",
 doi           = "10.1016/j.sysconle.2016.06.008",
 url           = "https://www.sciencedirect.com/science/article/pii/S0167691116300743",
 annote        = "This paper proposes lower bounds on a quantity called
                  $L^p$-norm joint spectral radius, or in short, $p$-radius,
                  of a finite set of matrices. Despite its wide range of
                  applications to, for example, stability analysis of
                  switched linear systems and the equilibrium analysis of
                  switched linear economical models, algorithms for computing
                  the $p$-radius are only available in a very limited number
                  of particular cases. The proposed lower bounds are given as
                  the spectral radius of an average of the given matrices
                  weighted via Kronecker products and do not place any
                  requirements on the set of matrices. We show that the
                  proposed lower bounds theoretically extend and also can
                  practically improve the existing lower bounds. A Markovian
                  extension of the proposed lower bounds is also presented.",
}

@MISC{TO:ArXiv09,
 author        = "Olivier, {\'E}ric and Thomas, Alain",
 title         = "Projective convergence of columns for inhomogeneous products
                  of matrices with nonnegative entries",
 howpublished  = "ArXiv.org e-Print archive",
 year          = "2009",
 month         = aug,
 eprinttype    = "arXiv",
 eprint        = "0908.4171",
 doi           = "10.48550/arXiv.0908.4171",
 annote        = "Let $P_n$ be the $n$-step right product $A_1\cdots A_n$,
                  where $A_1,A_2,\dots$ is a given infinite sequence of
                  $d\times d$ matrices with nonnegative entries. In a wide
                  range of situations, the normalized matrix product
                  $P_n/{\Vert P_n\Vert}$ does not converge and we shall be
                  rather interested in the asymptotic behavior of the
                  normalized columns $P_nU_i/\Vert P_nU_i\Vert$, where
                  $U_1,\dots,U_d$ are the canonical $d\times 1$ vectors. Our
                  main result in Theorem~A gives a sufficient condition ${\bf
                  (C)}$ over the sequence $A_1,A_2,\dots$ ensuring the
                  existence of {\it dominant columns} of $P_n$, having the
                  same projective limit $V$: more precisely, for any rank
                  $n$, there exists a partition of $\{1,\dots,d\}$ made of
                  two subsets $J_n\ne\emptyset$ and $J_n^c$ such that each
                  one of the sequences of normalized columns, say
                  $P_nU_{j_n}/\Vert P_nU_{j_n}\Vert$ with $j_n\in J_n$ tends
                  to $V$ as $n$ tends to $+\infty$ and are {\it dominant} in
                  the sense that the ratio $\Vert P_nU_{j_n'}/\Vert
                  P_nU_{j_n}\Vert$ tends to $0$, as soon as $j_n'\in J_n^c$.
                  The existence of sequences of such {\it dominant columns}
                  implies that for any probability vector $X$ with positive
                  entries, the probability vector $P_nX/\Vert P_nX\Vert$,
                  converges as $n$ tends to $+\infty$. Our main application
                  of Theorem~A (and our initial motivation) is related to an
                  {\it Erd\H os problem} concerned with a family of
                  probability measures $\mu_\beta$ (for $1<\beta<2$ a real
                  parameter) fully supported by a subinterval of the real
                  line, known as {\it Bernoulli convolutions}. For some
                  parameters $\beta$ (actually the so-called PV-numbers) such
                  measures are known to be {\it linearly representable}: the
                  $\mu_\beta$-measure of a suitable family of nested
                  generating intervals may be computed by means of matrix
                  products of the form $P_nX$, where $A_n$ takes only
                  finitely many values, say
                  $A(\mathtt{0}),\dots,A(\mathtt{a})$, and $X$ is a
                  probability vector with positive entries. Because,
                  $A_n=A(\xi_n)$, where $\xi=\xi_1\xi_2\cdots$ is a sequence
                  (one-sided infinite word) with
                  $\xi_n\in\{\mathtt{0},\dots,\mathtt{a}\}$, we shall write
                  $P_n=P_n(\xi)$ the dependence of the $n$-step product
                  with~$\xi$: when the convergence of ${P_n(\xi)X}/{\Vert
                  P_n(\xi)X\Vert}$ is uniform w.r.t. $\xi$, a sharp analysis
                  of the measure $\mu_\beta$ (Gibbs structures and
                  multifractal decomposition) becomes possible. However, most
                  of the matrices involved in the decomposition of these
                  Bernoulli convolutions are large, sparse and it is usually
                  not easy to prove the condition ${\bf (C)}$ of Theorem~A.
                  To illustrate the technics, we consider one parameter
                  $\beta$ for which the matrices are neither too small nor
                  too large and we verify condition ${\bf (C)}$: this leads
                  to the Gibbs properties of $\mu_\beta$.",
}

@ARTICLE{OmRadj:LAA97,
 author        = "Omladi{\v{c}}, Matja{\v{z}} and Radjavi, Heydar",
 title         = "Irreducible semigroups with multiplicative spectral radius",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "1997",
 volume        = "251",
 pages         = "59--72",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "47D03 (15A30 47A15)",
 mrnumber      = "1421265 (97k:47033)",
 mrreviewer    = "Leo Livshits",
 zblnumber     = "0937.47043",
 doi           = "10.1016/0024-3795(95)00544-7",
 url           = "https://www.sciencedirect.com/science/article/pii/0024379595005447",
 annote        = "The authors study irreducible multiplicative semigroups of
                  finite-dimensional linear operators with some additional
                  properties. They show that each of the properties --
                  submultiplicativity of the spectral radius, its
                  multiplicativity, or its constancy, as well as a certain
                  property called the Rota condition -- implies that every
                  member of such a semigroup is, except for a scalar
                  coefficient, similar to the direct sum of an isometry and a
                  nilpotent operator.",
}

@MASTERSTHESIS{Oreg-Reyes:MastThes18,
 author        = "Oreg{\'o}n-Reyes, Eduardo",
 authauthor    = "Eduardo Oreg{\'o}n-Reyes",
 title         = "Negative Curvature, Matrix Products, and Ergodic Theory",
 school        = "Faculty of Mathematics",
 address       = "Pontificia Universidad Cat{\'o}lica de Chile",
 year          = "2018",
 pagetotal     = "46",
 url           = "http://www.mat.uc.cl/~jairo.bochi/docs/Oregon-Reyes_master_thesis.pdf",
 annote        = "In the theses, some properties of sets of isometries of
                  Gromov hyperbolic spaces are investigated and a new
                  inequality about matrix products and a Berger--Wang formula
                  is obtained.",
}

@ARTICLE{Oreg-Reyes:GGD18,
 author        = "Oreg{\'o}n-Reyes, Eduardo",
 title         = "Properties of sets of isometries of {G}romov hyperbolic
                  spaces",
 journal       = "Groups Geom. Dyn.",
 fjournal      = "Groups, Geometry, and Dynamics",
 year          = "2018",
 volume        = "12",
 number        = "3",
 pages         = "889--910",
 eprinttype    = "arXiv",
 eprint        = "1606.01575",
 issn          = "1661-7207",
 mrclass       = "53C23 (20F65)",
 mrnumber      = "3844998",
 mrreviewer    = "Loreno Heer",
 zblnumber     = "1398.53049",
 doi           = "10.4171/GGD/468",
 url           = "https://dx.doi.org/10.4171/GGD/468",
 annote        = "We prove an inequality concerning isometries of a Gromov
                  hyperbolic metric space, which does not require the space
                  to be proper or geodesic. It involves the joint stable
                  length, a hyperbolic version of the joint spectral radius,
                  and shows that sets of isometries behave like sets of
                  $2\times 2$ real matrices. Among the consequences of the
                  inequality, we obtain the continuity of the joint stable
                  length and an analogue of Berger--Wang theorem.",
}

@ARTICLE{Oregon-Reyes:JEPM20,
 author        = "Oreg{\'o}n-Reyes, Eduardo",
 title         = "A new inequality about matrix products and a {B}erger-{W}ang
                  formula",
 journal       = "J. {\'E}c. polytech. Math.",
 fjournal      = "Journal de l'{\'E}cole polytechnique. Math{\'e}matiques",
 year          = "2020",
 volume        = "7",
 pages         = "185--200",
 eprinttype    = "arXiv",
 eprint        = "1710.00639",
 issn          = "2429-7100",
 mrclass       = "15A60 (15A42 37H15)",
 mrnumber      = "4054333",
 zblnumber     = "07152734",
 doi           = "10.5802/jep.114",
 url           = "https://jep.centre-mersenne.org/item/JEP_2020__7__185_0/",
 annote        = "We prove an inequality relating the norm of a product of
                  matrices $A_n\cdots A_1$ with the spectral radii of
                  subproducts $A_j\cdots A_i$ with $1\leq i\leq j\leq n$.
                  Among the consequences of this inequality, we obtain the
                  classical Berger--Wang formula as an immediate corollary,
                  and give an easier proof of a characterization of the upper
                  Lyapunov exponent due to I.~Morris. As main ingredient for
                  the proof of this result, we prove that for a large enough
                  $n$, the product $A_n\cdots A_1$ is zero under the
                  hypothesis that $A_j\cdots A_i$ are nilpotent for all
                  $1\leq i \leq j\leq n$. The main result is the
                  following:\par \textbf{Theorem.} \emph{Given $d \in
                  \mathbb{N}$, there exists an integer $N\geq 1$, and a
                  constant $0<\delta<1$ such that, for every local field $k$
                  and norm $\|\cdot\|$ in $M_{d}(k)$, there is a constant
                  $C=C(k,d,\|\cdot\|)>1$ satisfying the following inequality
                  for all $A_1,\dots,A_N \in M_{d}(k)$:} \[ \|A_N\cdots A_1\|
                  \leq C \left(\prod_{1\leq i\leq
                  N}{\|A_i\|}\right)\max_{1\leq \alpha\leq \beta\leq N}{
                  \left( \frac{\rho(A_{\beta}\cdots A_{\alpha}) }{
                  \prod_{\alpha\leq i\leq \beta}{\|A_i\|}} \right)^{\delta}},
                  \] \emph{where the right hand side is treated as zero if
                  one of the $A_{i}$ is the zero matrix.}",
}

@ARTICLE{PanScl:Nonlin21,
 author        = "Panti, Giovanni and Sclosa, Davide",
 title         = "The finiteness conjecture holds in
                  {$(\mathrm{SL}_2\mathbb{Z}_{\ge0})^2$}",
 journal       = "Nonlinearity",
 fjournal      = "Nonlinearity",
 year          = "2021",
 volume        = "34",
 number        = "8",
 pages         = "5234--5260",
 eprinttype    = "arXiv",
 eprint        = "2006.16899",
 issn          = "0951-7715",
 mrclass       = "15A60 (05A05 37F32)",
 mrnumber      = "4281474",
 mrreviewer    = "Oliver Patrick Mason",
 zblnumber     = "1467.05008",
 doi           = "10.1088/1361-6544/ac0484",
 url           = "https://iopscience.iop.org/article/10.1088/1361-6544/ac0484",
 annote        = "Let $A,B$ be matrices in $\mathrm{SL}_2\mathbb{R}$ having
                  trace greater than or equal to~$2$. Assume the pair $A,B$
                  is coherently oriented, that is, can be conjugated to a
                  pair having nonnegative entries. Assume also that either
                  $A,B^{-1}$ is coherently oriented as well, or $A,B$ have
                  integer entries. Then the Lagarias--Wang finiteness
                  conjecture holds for the set $\{A,B\}$, with optimal
                  product in $\{A,B,AB,A^2B,AB^2\}$. In particular, it holds
                  for every matrix pair in
                  $\mathrm{SL}_2\mathbb{Z}_{\ge0}$.",
}

@INCOLLECTION{ParJdb:LNCS07,
 author        = "Parrilo, Pablo A. and Jadbabaie, Ali",
 authauthor    = "Parrilo, Pablo and Jadbabaie, Ali",
 title         = "Approximation of the joint spectral radius of a set of
                  matrices using sum of squares",
 booktitle     = "Hybrid systems: computation and control",
 series        = "Lecture Notes in Comput. Sci.",
 publisher     = "Springer",
 address       = "Berlin",
 year          = "2007",
 volume        = "4416",
 pages         = "444--458",
 mrclass       = "15A18 (93D05)",
 mrnumber      = "2363632 (2008m:15027)",
 mrreviewer    = "Oleg N. Kirillov",
 zblnumber     = "1221.65098",
 doi           = "10.1007/978-3-540-71493-4_35",
 url           = "https://link.springer.com/chapter/10.1007/978-3-540-71493-4_35",
 annote        = "We provide an asymptotically tight, computationally
                  efficient approximation of the joint spectral radius of a
                  set of matrices using sum of squares (SOS) programming. The
                  approach is based on a search for a SOS polynomial that
                  proves simultaneous contractibility of a finite set of
                  matrices. We provide a bound on the quality of the
                  approximation that unifies several earlier results and is
                  independent of the number of matrices. Additionally, we
                  present a comparison between our approximation scheme and a
                  recent technique due to Blondel and Nesterov, based on
                  lifting of matrices. Theoretical results and numerical
                  investigations show that our approach yields tighter
                  approximations.",
}

@ARTICLE{ParJdb:LAA08,
 author        = "Parrilo, Pablo A. and Jadbabaie, Ali",
 authauthor    = "Parrilo, Pablo and Jadbabaie, Ali",
 title         = "Approximation of the joint spectral radius using sum of
                  squares",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2008",
 volume        = "428",
 number        = "10",
 pages         = "2385--2402",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15A18 (47A12 47A13)",
 mrnumber      = "2408034 (2009e:15033)",
 mrreviewer    = "Nicola Guglielmi",
 zblnumber     = "1151.65032",
 doi           = "10.1016/j.laa.2007.12.027",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379508000281",
 annote        = "We provide an asymptotically tight, computationally
                  efficient approximation of the joint spectral radius of a
                  set of matrices using sum of squares (SOS) programming. The
                  approach is based on a search for an SOS polynomial that
                  proves simultaneous contractibility of a finite set of
                  matrices. We provide a bound on the quality of the
                  approximation that unifies several earlier results and is
                  independent of the number of matrices. Additionally, we
                  present a comparison between our approximation scheme and
                  earlier techniques, including the use of common quadratic
                  Lyapunov functions and a method based on matrix liftings.
                  Theoretical results and numerical investigations show that
                  our approach yields tighter approximations.",
}

@ARTICLE{Pascoe:IJM21,
 author        = "Pascoe, James E.",
 title         = "The outer spectral radius and dynamics of completely
                  positive maps",
 journal       = "Israel J. Math.",
 fjournal      = "Israel Journal of Mathematics",
 year          = "2021",
 volume        = "244",
 number        = "2",
 pages         = "945--969",
 eprinttype    = "arXiv",
 eprint        = "1905.09895",
 issn          = "0021-2172",
 mrclass       = "15A60 (15B48 37A55 81P47)",
 mrnumber      = "4344050",
 zblnumber     = "07436566",
 doi           = "10.1007/s11856-021-2198-x",
 url           = "https://link.springer.com/article/10.1007/s11856-021-2198-x",
 annote        = "We examine a special case of an approximation of the joint
                  spectral radius given by Blondel and Nesterov, which we
                  call the outer spectral radius. The outer spectral radius
                  is given by the square root of the ordinary spectral radius
                  of the $n^2$ by $n^2$ matrix $\sum\bar{X}_i\otimes X_i$. We
                  give an analogue of the spectral radius formula for the
                  outer spectral radius which can be used to quickly obtain
                  the error bounds in methods based on the work of Blondel
                  and Nesterov. The outer spectral radius is used to analyze
                  the iterates of a completely positive map, including the
                  special case of quantum channels. The average of the
                  iterates of a completely positive map approach to a
                  completely positive map where the Kraus operators span an
                  ideal in the algebra generated by the Kraus operators of
                  the original completely positive map. We also give an
                  elementary treatment of Popescu's theorems on similarity to
                  row contractions in the matrix case, describe connections
                  to the Parrilo--Jadbabaie relaxation, and give a detailed
                  analysis of the maximal spectrum of a completely positive
                  map.",
}

@ARTICLE{Pep:LAA08,
 author        = "Peperko, Aljo{\v{s}}a",
 title         = "On the max version of the generalized spectral radius
                  theorem",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2008",
 volume        = "428",
 number        = "10",
 pages         = "2312--2318",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15A18 (16Y60 47A13)",
 mrnumber      = "2405247 (2009a:15035)",
 mrreviewer    = "Ross A. Lippert",
 zblnumber     = "1144.15006",
 zblreviewer   = "Sheng Chen (Harbin)",
 doi           = "10.1016/j.laa.2007.08.041",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379507003898",
 annote        = "The generalized spectral radius theorem states that the
                  generalized spectral radius and the joint spectral radius
                  of a bounded set of square complex matrices coincide. A max
                  algebra version of the generalized spectral radius theorem
                  was proposed by \emph{Y.-Y.~Lur}~\cite{Lur:LAA06-1}. This
                  paper gives a short proof of it by generalizing a result
                  about the relationship between the maximum circuit
                  geometric mean and Hadamard powers of a matrix, which was
                  proved by \emph{S.~Friedland} [Linear Algebra Appl. 74,
                  173--178 (1986)].",
}

@ARTICLE{Pep:LAA11,
 author        = "Peperko, Aljo{\v{s}}a",
 title         = "On the continuity of the generalized spectral radius in max
                  algebra",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2011",
 volume        = "435",
 number        = "4",
 pages         = "902--907",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15A80 (15A18 15A60)",
 mrnumber      = "2807242 (2012d:15057)",
 mrreviewer    = "Buket Benek Gursoy",
 zblnumber     = "1221.15020",
 zblreviewer   = "Mihail Voicu (Ia{\c{s}}i)",
 doi           = "10.1016/j.laa.2011.02.015",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379511001261",
 annote        = "Given a bounded set $\Psi$ of $n\times n$ non-negative
                  matrices, let $\rho(\Psi)$ and $\mu(\Psi)$ denote the
                  generalized spectral radius of $\Psi$ and its max version,
                  respectively. We show that \[\mu(\Psi)=
                  \sup_{t\in(0,\infty)}\left(n^{-1}\rho(\Psi^{(t)})\right)^{1/t},\]
                  where $\Psi^{(t)}$ denotes the Hadamard power of $\Psi$. We
                  apply this result to give a new short proof of a known fact
                  that $\mu(\Psi)$ is continuous on the Hausdorff metric
                  space $(\beta,H)$ of all nonempty compact collections of
                  $n\times n$ non-negative matrices.",
}

@ARTICLE{Pep:LMA11,
 author        = "Peperko, Aljo{\v{s}}a",
 title         = "On the functional inequality for the spectral radius of
                  compact operators",
 journal       = "Linear Multilinear Algebra",
 fjournal      = "Linear and Multilinear Algebra",
 year          = "2011",
 volume        = "59",
 number        = "4",
 pages         = "357--364",
 issn          = "0308-1087",
 mrclass       = "47A12 (47B37)",
 mrnumber      = "2802518 (2012c:47017)",
 mrreviewer    = "Arkady K. Kitover",
 zblnumber     = "1223.47007",
 zblreviewer   = "{\c{S}}afak Alpay (Ankara)",
 doi           = "10.1080/03081080903489985",
 url           = "https://www.tandfonline.com/doi/abs/10.1080/03081080903489985",
 annote        = "Let $A_1,A_2,\ldots,A_n$ be matrices of the same order and
                  let $r$ denote the spectral radius. Functions
                  $F:\mathbb{R}_{+}^{n}\to\mathbb{R}$ satisfying
                  \[r(F(A_1,A_2,\ldots,A_n))\le
                  F(r(A_1),r(A_2),\ldots,r(A_n))\] for all non-negative
                  matrices $A_1,A_2,\ldots,A_n$ were characterized by
                  \emph{L.~Elsner, D.~Hershkowitz} and \emph{A.~Pincus}
                  [Linear Algebra Appl. 129, 103--130 (1990)]. The author
                  generalizes this result to the setting of infinite
                  non-negative matrices that define compact operators on a
                  Banach sequence space.",
}

@ARTICLE{Peperko:LAA12,
 author        = "Peperko, Aljo{\v{s}}a",
 title         = "Bounds on the generalized and the joint spectral radius of
                  {H}adamard products of bounded sets of positive operators
                  on sequence spaces",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2012",
 volume        = "437",
 number        = "1",
 pages         = "189--201",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15A60 (15A80 15B48)",
 mrnumber      = "2917439",
 mrreviewer    = "Borut Zalar",
 zblnumber     = "1243.15011",
 zblreviewer   = "Mohammad Sal Moslehian (Mashhad)",
 doi           = "10.1016/j.laa.2012.02.022",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379512001681",
 annote        = "Suppose that $A,B,A_1,\ldots,A_m$ are nonnegative $n\times
                  n$ matrices, and $\circ$ and $\rho$ denote the Hadamard
                  product and the spectral radius, respectively. It is a
                  known fact that the spectral radius of the Hadamard product
                  of two non-negative matrices is submultiplicative.
                  \emph{K.\,M.\,R.~Audenaert} [Linear Algebra Appl. 432, No.
                  1, 366--368 (2010)] proved a conjecture of \emph{X.~Zhan}
                  [Advanced Workshop on Trends and Developments in Linear
                  Algebra, ICTP, Trieste (2009)] by establishing that
                  \[\rho(A\circ B)\le\rho^{1/2}((A\circ A)(B\circ
                  B))\le\rho(AB).\] \emph{R.\,A.~Horn} and \emph{F.~Zhang}
                  [Electron. J. Linear Algebra 20, 90--94 (2010)] showed that
                  \[\rho(A\circ B)\le\rho^{1/2}(AB\circ BA)\le\rho(AB).\]
                  \emph{Z.~Huang} [Linear Algebra Appl. 434, No. 2, 457--462
                  (2011)] proved that $\rho(A_1\circ\cdots\circ A_m)\le
                  \rho(A_1 \cdots A_m)$.\par In the paper the author extends
                  the results above to the setting of the generalized and the
                  joint spectral radius of bounded sets of non-negative
                  matrices that define bounded operators on Banach sequence
                  spaces. He also proves the inequalities \[\rho(A\circ
                  B)\le\rho^{1/2}((A\circ A)(B\circ B))\le \rho(AB\circ
                  AB)^{1/4} \rho(BA\circ BA)^{1/4}\le \rho(AB)\] and
                  \[\rho(A\circ B)\le\rho^{1/2}(AB\circ BA)\le \rho(AB\circ
                  AB)^{1/4}\rho(BA\circ BA)^{1/4}\le \rho(AB)\] in the case
                  of the usual spectral radius of non-negative matrices.",
}

@ARTICLE{PEDJ:Autom16,
 author        = "Philippe, Matthew and Essick, Ray and Dullerud, Geir E. and
                  Jungers, Rapha{\"e}l M.",
 authauthor    = "Philippe, Matthew and Essick, Ray and Dullerud, Geir and
                  Jungers, Rapha{\"e}l",
 title         = "Stability of discrete-time switching systems with
                  constrained switching sequences",
 journal       = "Automatica J. IFAC",
 year          = "2016",
 volume        = "72",
 pages         = "242--250",
 eprinttype    = "arXiv",
 eprint        = "1503.06984",
 issn          = "0005-1098",
 mrclass       = "93D05 (93D30)",
 mrnumber      = "3542938",
 mrreviewer    = "John Tsinias",
 zblnumber     = "1344.93088",
 doi           = "10.1016/j.automatica.2016.05.015",
 url           = "https://www.sciencedirect.com/science/article/pii/S000510981630200X",
 annote        = "We introduce a novel framework for the stability analysis of
                  discrete-time linear switching systems with switching
                  sequences constrained by an automaton. The key element of
                  the framework is the algebraic concept of multinorm, which
                  associates a different norm per node of the automaton, and
                  allows to exactly characterize stability. Building upon
                  this tool, we develop the first arbitrarily accurate
                  approximation schemes for estimating the constrained joint
                  spectral radius $\hat{\rho}$, that is the exponential
                  growth rate of a switching system with constrained
                  switching sequences. More precisely, given a relative
                  accuracy $r>0$, the algorithms compute an estimate of
                  $\hat{\rho}$ within the range $[\hat{\rho},
                  (1+r)\hat{\rho}]$. These algorithms amount to solve a well
                  defined convex optimization program with known
                  time-complexity, and whose size depends on the desired
                  relative accuracy $r>0$.",
}

@INPROCEEDINGS{PhilJung:ECC15,
 author        = "Philippe, Matthew and Jungers, Rapha{\"e}l M.",
 authauthor    = "Philippe, Matthew and Jungers, Rapha{\"e}l",
 title         = "Converse {L}yapunov theorems for discrete-time linear
                  switching systems with regular switching sequences",
 booktitle     = "Proceedings of the European Control Conference (ECC), 15-17
                  July 2015",
 publisher     = "IEEE",
 address       = "Linz, Austria",
 year          = "2015",
 eprinttype    = "arXiv",
 eprint        = "1410.7197",
 doi           = "10.1109/ECC.2015.7330816",
 url           = "https://ieeexplore.ieee.org/document/7330816",
 annote        = "We present a stability analysis framework for the general
                  class of discrete-time linear switching systems for which
                  the switching sequences belong to a regular language. They
                  admit arbitrary switching systems as special cases.\par
                  Using recent results of \emph{X.~Dai}~\cite{Dai:LAA12} on
                  the asymptotic growth rate of such systems, we introduce
                  the concept of \emph{multinorm} as an algebraic tool for
                  stability analysis.\par We conjugate this tool with two
                  families of multiple quadratic Lyapunov functions,
                  parameterized by an integer $T \geq 1$, and obtain converse
                  Lyapunov Theorems for each.\par Lyapunov functions of the
                  first family associate one quadratic form per state of the
                  automaton defining the switching sequences. They are made
                  to decrease after every $T$ successive time steps. The
                  second family is made of the \textit{path-dependent}
                  Lyapunov functions of Lee and Dullerud. They are
                  parameterized by an amount of memory $(T-1) \geq 0$.\par
                  Our converse Lyapunov theorems are finite. More precisely,
                  we give sufficient conditions on the asymptotic growth rate
                  of a stable system under which one can compute an integer
                  parameter $T \geq 1$ for which both types of Lyapunov
                  functions exist. As a corollary of our results, we
                  formulate an arbitrary accurate approximation scheme for
                  estimating the asymptotic growth rate of switching systems
                  with constrained switching sequences.",
}

@INPROCEEDINGS{PhilJung:ICHS15,
 author        = "Philippe, Matthew and Jungers, Rapha{\"e}l M.",
 authauthor    = "Philippe, Matthew and Jungers, Rapha{\"e}l",
 title         = "A Sufficient Condition for the Boundedness of Matrix
                  Products Accepted by an Automaton",
 booktitle     = "Proceedings of the 18th International Conference on Hybrid
                  Systems: Computation and Control",
 series        = "HSCC'15",
 publisher     = "ACM",
 address       = "New York, NY, USA",
 location      = "Seattle, Washington",
 year          = "2015",
 pages         = "51--57",
 pagetotal     = "7",
 isbn          = "978-1-4503-3433-4",
 mrclass       = "15A30 (68Q45 93D09)",
 mrnumber      = "3440811",
 zblnumber     = "1366.68151",
 acmid         = "2728610",
 doi           = "10.1145/2728606.2728610",
 url           = "https://dl.acm.org/doi/10.1145/2728606.2728610",
 annote        = "We study the boundedness of products of matrices associated
                  with words in a regular language. This question naturally
                  arises in the stability analysis of switching systems with
                  constrained switching sequences.\par Our main contribution
                  is a sufficient condition for the boundedness of all the
                  possible products of matrices that may occur in a
                  marginally unstable system. We show that this condition can
                  be expressed in terms of products of finite lengths, and is
                  therefore algorithmically checkable.\par We then compare
                  our condition with a second one, inspired by a lifting
                  procedure introduced by Kozyakin, and prove that our
                  condition is at least as powerful as this second one.",
}

@ARTICLE{PW:LAA08,
 author        = "Plischke, Elmar and Wirth, Fabian",
 title         = "Duality results for the joint spectral radius and transient
                  behavior",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2008",
 volume        = "428",
 number        = "10",
 pages         = "2368--2384",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15A18 (15A60 47A13)",
 mrnumber      = "2408033 (2009g:15033)",
 mrreviewer    = "Nicola Mastronardi",
 zblnumber     = "05268361",
 doi           = "10.1016/j.laa.2007.12.009",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379507005691",
 annote        = "For linear inclusions in discrete or continuous time several
                  quantities characterizing the growth behavior of the
                  corresponding semigroup are analyzed. These quantities are
                  the joint spectral radius, the initial growth rate and (for
                  bounded semigroups) the transient bound. It is discussed
                  how these constants relate to one another and how they are
                  characterized by various norms. A complete duality theory
                  is developed in this framework, relating semigroups and
                  dual semigroups and extremal or transient norms with their
                  respective dual norms.",
}

@INPROCEEDINGS{PWB:CDC05,
 author        = "Plischke, Elmar and Wirth, Fabian and Barabanov, Nikita",
 title         = "Duality Results for the Joint Spectral Radius and Transient
                  Behavior",
 booktitle     = "Proceedings of the 44th {IEEE} Conference on Decision and
                  Control and European Control Conference 2005, Seville,
                  Spain, December 12--15",
 year          = "2005",
 pages         = "2344--2349",
 doi           = "10.1109/CDC.2005.1582512",
 annote        = "For linear inclusions in discrete or continuous time several
                  quantities characterizing the growth behavior of the
                  corresponding semigroup are analyzed. These quantities are
                  the joint spectral radius, the initial growth rate and (for
                  bounded semigroups) the transient bound. It is recalled how
                  these constants relate to one another and how they are
                  characterized by various norms. A complete duality theory
                  is developed in this framework, relating semigroups and
                  dual semigroups and extremal or transient norms with their
                  respective dual norms.",
}

@ARTICLE{Prot:FPM96:e,
 author        = "Protasov, V. {\relax{}Yu}.",
 authauthor    = "Protasov, V.",
 title         = "The joint spectral radius and invariant sets of linear
                  operators",
 journal       = "Fundam. Prikl. Mat.",
 fjournal      = "Fundamental'naya i Prikladnaya Matematika",
 year          = "1996",
 volume        = "2",
 number        = "1",
 pages         = "205--231",
 issn          = "1560-5159",
 mrclass       = "47A13 (15A60)",
 mrnumber      = "1789006 (2002d:47006)",
 mrreviewer    = "Zden{\v{e}}k Dost{\'a}l",
 zblnumber     = "0899.47002",
 note          = "In Russian",
 annote        = "This paper concerns the properties of the joint spectral
                  radius of several linear $n$-dimensional operators:
                  \[\widehat{\rho} (A_1,\ldots, A_k)= \lim_{m\to\infty}
                  \max_\sigma \|A_{\sigma(1)}\cdots
                  A_{\sigma(m)}\|^{\frac{1}{m}}, \quad \sigma: \{1,\ldots,
                  m\}\to \{1,\ldots, k\}.\] The theorem of
                  Dranishnikov--Konyagin on the existence of an invariant
                  convex set $M$ for several linear operators is proved.
                  $\text{Conv} (A_1 M,\ldots, A_k M)=\lambda M$, $\lambda=
                  \widehat{\rho} (A_1,\ldots, A_k)$. \par The paper concludes
                  with several boundary propositions on the construction of
                  invariant sets some properties of the invariant sets and
                  algorithm of finding the joint spectral radius with
                  estimation of its difficulty.",
}

@ARTICLE{Prot:IZV97:e,
 author        = "Protasov, V. {\relax{}Yu}.",
 authauthor    = "Protasov, V.",
 title         = "A generalized joint spectral radius. {A} geometric
                  approach",
 journal       = "Izv. Math.",
 fjournal      = "Izvestiya: Mathematics",
 year          = "1997",
 volume        = "61",
 number        = "5",
 pages         = "995--1030",
 issn          = "1064-5632",
 mrclass       = "15A48 (47A13 65F99)",
 mrnumber      = "1486700 (99c:15041)",
 mrreviewer    = "L. Rodman",
 zblnumber     = "0893.15002",
 doi           = "10.1070/im1997v061n05ABEH000161",
 url           = "https://iopscience.iop.org/article/10.1070/IM1997v061n05ABEH000161",
 annote        = "The properties of the joint spectral radius with an
                  arbitrary exponent $p\in [1,+\infty]$ are investigated for
                  a set of finite-dimensional linear operators
                  $A_1,\ldots,A_k$, \[\displaystyle\widehat\rho_p
                  =\lim_{n\to\infty} \biggl(\dfrac{1}{k^n}\,\sum_{\sigma}
                  \|A_{\sigma(1)}\cdots
                  A_{\sigma(n)}\|^p\biggr)^{\frac{1}{pn}},\quad p<\infty,\]
                  \[\displaystyle\widehat\rho_{\infty} =
                  \lim_{n\to\infty}\max_{\sigma} \|A_{\sigma(1)}\cdots
                  A_{\sigma(n)}\|^{\frac{1}{n}},\] where the summation and
                  maximum extend over all maps \[\sigma
                  \colon\{1,\ldots,n\}\to\{1,\ldots,k\}.\] Using the
                  operation of generalized addition of convex sets, we extend
                  the Dranishnikov--Konyagin theorem on invariant convex
                  bodies, which has hitherto been established only for the
                  case $p=\infty$. The paper concludes with some assertions
                  on the properties of invariant bodies and their
                  relationship to the spectral radius $\widehat\rho_p$. The
                  problem of calculating $\widehat\rho_p$ for even integers
                  $p$ is reduced to determining the usual spectral radius for
                  an appropriate finite-dimensional operator. For other
                  values of $p$, a geometric analogue of the method with a
                  pre-assigned accuracy is constructed and its complexity is
                  estimated.",
}

@ARTICLE{Prot:SM00,
 author        = "Protasov, V. {\relax{}Yu}.",
 authauthor    = "Protasov, V.",
 title         = "Asymptotics of the partition function",
 journal       = "Mat. Sb.",
 fjournal      = "Matematicheski{\u\i} Sbornik",
 year          = "2000",
 volume        = "191",
 number        = "3",
 pages         = "65--98",
 issn          = "0368-8666",
 mrclass       = "11P82 (11N56)",
 mrnumber      = "1773255",
 mrreviewer    = "Angel V. Kumchev",
 zblnumber     = "1004.11056",
 doi           = "10.1070/sm2000v191n03ABEH000464",
 url           = "https://iopscience.iop.org/article/10.1070/SM2000v191n03ABEH000464",
 annote        = "Some estimates for the joint and the lower spectral radii of
                  the operators having a common invariant cone are obtained
                  which then used in investigation of the following
                  problem.\par Given a pair of positive integers $m$ and $d$
                  such that $2\leqslant m\leqslant d$, for integer
                  $n\geqslant 0$ the quantity $b_{m,d}(n)$, called the
                  partition function is considered; this by definition is
                  equal to the cardinality of the set \[
                  \biggl\{(a_0,a_1,\dots):n=\sum_ka_km^k,\
                  a_k\in\{0,\dots,d-1\},\ k\geqslant 0\biggr\}. \] The
                  properties of $b_{m,d}(n)$ and its asymptotic behaviour as
                  $n\to\infty$ are studied. A geometric approach to this
                  problem is put forward. It is shown that
                  \[C_1n^{\lambda_1}\leqslant b_{m,d}(n)\leqslant
                  C_2n^{\lambda_2},\] for sufficiently large $n$ , where
                  $C_1$ and $C_2$ are positive constants depending on $m$ and
                  $d$, and
                  $\lambda_1=\varliminf\limits_{n\to\infty}\dfrac{\log
                  b(n)}{\log n}$ and
                  $\lambda_2=\varlimsup\limits_{n\to\infty}\dfrac{\log
                  b(n)}{\log n}$ are characteristics of the exponential
                  growth of the partition function. For some pair $(m,d)$ the
                  exponents $\lambda_1$ and $\lambda_2$ are calculated as the
                  logarithms of certain algebraic numbers; for other pairs
                  the problem is reduced to finding the joint spectral radius
                  of a suitable collection of finite-dimensional linear
                  operators. Estimates of the growth exponents and the
                  constants $C_1$and $C_2$ are obtained.",
}

@ARTICLE{Prot:LAA08,
 author        = "Protasov, V. {\relax{}Yu}.",
 authauthor    = "Protasov, V.",
 title         = "Extremal {$L_{p}$}-norms of linear operators and
                  self-similar functions",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2008",
 volume        = "428",
 number        = "10",
 pages         = "2339--2356",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "47A30 (26A16 28A80 39B22 47B38)",
 mrnumber      = "2408031 (2009a:47018)",
 mrreviewer    = "Bin Han",
 zblnumber     = "1147.15023",
 zblreviewer   = "Omar Hirzallah (Zarqa)",
 doi           = "10.1016/j.laa.2007.09.023",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379507004387",
 annote        = "We prove that for any $p\in [0,\infty]$ a finite irreducible
                  family of linear operators possesses an extremal norm
                  corresponding to the $p$-radius of these operators. As a
                  corollary, we derive a criterion for the
                  $L_{p}$-contractibility property of linear operators and
                  estimate the asymptotic growth of orbits for any point.
                  These results are applied to the study of functional
                  difference equations with linear contractions of the
                  argument (self-similarity equations). We obtain a sharp
                  criterion for the existence and uniqueness of solutions in
                  various functional spaces, compute the exponents of
                  regularity, and estimate moduli of continuity. This, in
                  particular, gives a geometric interpretation of the
                  $p$-radius in terms of spectral radii of certain operators
                  in the space $L_{p}[0,1]$.",
}

@ARTICLE{Prot:RMS10,
 author        = "Protasov, V. {\relax{}Yu}.",
 authauthor    = "Protasov, V.",
 title         = "Semigroups of nonnegative matrices",
 journal       = "Russ. Math. Surv.",
 fjournal      = "Russian Mathematical Surveys",
 year          = "2010",
 volume        = "65",
 number        = "6",
 pages         = "1186--1188",
 issn          = "0036-0279",
 mrclass       = "20M20",
 mrnumber      = "2779367 (2012b:20152)",
 mrreviewer    = "Inessa Levi",
 zblnumber     = "1217.20037",
 doi           = "10.1070/RM2010v065n06ABEH004722",
 url           = "https://iopscience.iop.org/article/10.1070/RM2010v065n06ABEH004722",
 annote        = "Let $\mathcal{S}$ be an arbitrary multiplicative semigroup
                  of $n\times n$ matrices, all of which are non-negative,
                  that is, have non-negative entries. In particular,
                  $\mathcal{S}$ can be defined as the set of all products of
                  a given family of non-negative matrices. Under what
                  conditions does $\mathcal{S}$ contain a strictly positive
                  matrix? In the special case when $\mathcal{S}$ is generated
                  by one matrix, that is, $\mathcal{S}=\{A^{k},
                  k\in\mathbb{N}\}$, several combinatorial and analytical
                  criteria are known. In addition, for this case there are
                  many results on the minimal positive power of the matrix
                  and on the general structure of matrices that have no
                  positive powers. In this paper we derive a criterion for
                  the existence of a positive element in a given matrix
                  semigroup (Proposition 1). As a corollary we prove the
                  polynomial solvability of this problem for semigroups
                  generated by finite families of matrices (Theorem 1).",
}

@ARTICLE{Prot:MS11,
 author        = "Protasov, V. {\relax{}Yu}.",
 authauthor    = "Protasov, V.",
 title         = "Invariant functions for the {L}yapunov exponents of random
                  matrices",
 journal       = "Sb. Math.",
 fjournal      = "Sbornik: Mathematics",
 year          = "2011",
 volume        = "202",
 number        = "1",
 pages         = "101--126",
 issn          = "1064-5616",
 mrclass       = "60B20 (15B52)",
 mrnumber      = "2796828 (2012f:60029)",
 mrreviewer    = "Oleksiy Khorunzhiy",
 zblnumber     = "1239.60004",
 zblreviewer   = "Paul-Olivier Dehaye (Z{\"u}rich)",
 doi           = "10.1070/SM2011v202n01ABEH004139",
 url           = "https://iopscience.iop.org/article/10.1070/SM2011v202n01ABEH004139",
 annote        = "A new approach to the study of Lyapunov exponents of random
                  matrices is presented. We prove that any family of
                  nonnegative $(d\times d)$-matrices has a continuous concave
                  invariant functional on $\mathbb{R}^{d}_{+}$. Under some
                  standard assumptions on the matrices, this functional is
                  strictly positive, and the coefficient corresponding to it
                  is equal to the largest Lyapunov exponent. As a corollary
                  we obtain asymptotics for the expected value of the
                  logarithm of norms of matrix products and of their spectral
                  radii. Another corollary gives new upper and lower bounds
                  for the Lyapunov exponent, and an algorithm for computing
                  it for families of nonnegative matrices. We consider
                  possible extensions of our results to general nonnegative
                  matrix families and present several applications and
                  examples.",
}

@INPROCEEDINGS{PJ:CDC13,
 author        = "Protasov, V. {\relax{}Yu}. and Jungers, R. M.",
 authauthor    = "Protasov, V. and Jungers, R.",
 title         = "Is switching systems stability harder for continuous time
                  systems?",
 booktitle     = "Proceedings of the 52nd IEEE Conference on Decision and
                  Control (CDC), 10--13 Dec.",
 publisher     = "IEEE",
 address       = "Florence, Italy",
 year          = "2013",
 pages         = "704--709",
 doi           = "10.1109/CDC.2013.6759964",
 url           = "https://ieeexplore.ieee.org/document/6759964",
 annote        = "We analyse the problem of stability of a continuous time
                  linear switching system (LSS) versus the stability of its
                  Euler discretization. It is well-known that the existence
                  of a positive $\tau$ for which the corresponding discrete
                  time system with stepsize $\tau$ is stable implies the
                  stability of the LSS. Our main goal is to obtain a converse
                  statement, that is to estimate the discretization stepsize
                  $\tau>0$ up to a given accuracy $\varepsilon>0$. This would
                  lead to a method for deciding the stability of a continuous
                  time LSS with a guaranteed accuracy. As a first step
                  towards the solution of this problem, we show that for
                  systems of matrices with real spectrum the parameter $\tau$
                  can be effectively estimated. We prove that in this special
                  case, the discretized system is stable if and only if the
                  Lyapunov exponent of the LSS is smaller than $-C\tau$,
                  where $C$ is an effective constant depending on the system.
                  The proofs are based on applying Markov--Bernstein type
                  inequalities for systems of exponents.",
}

@ARTICLE{PJ:LAA13,
 author        = "Protasov, V. {\relax{}Yu}. and Jungers, R. M.",
 authauthor    = "Protasov, V. and Jungers, R.",
 title         = "Lower and upper bounds for the largest {L}yapunov exponent
                  of matrices",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2013",
 volume        = "438",
 number        = "11",
 pages         = "4448--4468",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "60H25 (15A15 15A60)",
 mrnumber      = "3034543",
 zblnumber     = "1281.65154",
 doi           = "10.1016/j.laa.2013.01.027",
 url           = "https://www.sciencedirect.com/science/article/pii/S002437951300089X",
 annote        = "We introduce a new approach to evaluate the largest Lyapunov
                  exponent of a family of nonnegative matrices. The method is
                  based on using special positive homogeneous functionals on
                  $\mathbb{R}^{d}_{+}$, which gives iterative lower and upper
                  bounds for the Lyapunov exponent. They improve previously
                  known bounds and converge to the real value. The rate of
                  convergence is estimated and the efficiency of the
                  algorithm is demonstrated on several problems from
                  applications (in functional analysis, combinatorics, and
                  language theory) and on numerical examples with randomly
                  generated matrices. The method computes the Lyapunov
                  exponent with a prescribed accuracy in relatively high
                  dimensions (up to $60$). We generalize this approach to all
                  matrices, not necessarily nonnegative, derive a new
                  universal upper bound for the Lyapunov exponent, and show
                  that a potential similar lower bound does not exist in
                  general.",
}

@ARTICLE{ProtVoyn:LAA12,
 author        = "Protasov, V. {\relax{}Yu}. and Voynov, A. S.",
 authauthor    = "Protasov, V. and Voynov, A.",
 title         = "Sets of nonnegative matrices without positive products",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2012",
 volume        = "437",
 number        = "3",
 pages         = "749--765",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15B48 (05A05)",
 mrnumber      = "2921734",
 mrreviewer    = "Daniel B. Szyld",
 zblnumber     = "1245.15033",
 zblreviewer   = "K. C. Sivakumar (Chennai)",
 doi           = "10.1016/j.laa.2012.02.029",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379512002005",
 annote        = "For an arbitrary irreducible set of nonnegative $d\times
                  d$-matrices, we consider the following problem: does there
                  exist a strictly positive product (with repetitions
                  permitted) of those matrices? Under some general
                  assumptions, we prove that if it does not exist, then there
                  is a partition of the set of basis vectors of
                  $\mathbb{R}^d$, on which all given matrices act as
                  permutations. Moreover, there always exists a unique
                  maximal partition (with the maximal number of parts)
                  possessing this property, and the number of parts is
                  expressed by eigenvalues of matrices. This generalizes
                  well-known results of Perron--Frobenius theory on
                  primitivity of one matrix to families of matrices. We
                  present a polynomial algorithm to decide the existence of a
                  positive product for a given finite set of matrices and to
                  build the maximal partition. Similar results are obtained
                  for scrambling products. Applications to the study of
                  Lyapunov exponents, inhomogeneous Markov chains, etc. are
                  discussed.",
}

@ARTICLE{ProtVoy:LAA17,
 author        = "Protasov, V. {\relax{}Yu}. and Voynov, A. S.",
 authauthor    = "Protasov, V. and Voynov, A.",
 title         = "Matrix semigroups with constant spectral radius",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2017",
 volume        = "513",
 pages         = "376--408",
 eprinttype    = "arXiv",
 eprint        = "1407.6568",
 issn          = "0024-3795",
 mrclass       = "47D03 (15A30 15A60 15B36)",
 mrnumber      = "3573807",
 zblnumber     = "1359.15010",
 doi           = "10.1016/j.laa.2016.10.013",
 url           = "https://www.sciencedirect.com/science/article/abs/pii/S0024379516304864",
 annote        = "Multiplicative matrix semigroups with constant spectral
                  radius (c.s.r.) are studied and applied to several problems
                  of algebra, combinatorics, functional equations, and
                  dynamical systems. We show that all such semigroups are
                  characterized by means of irreducible ones. Each
                  irreducible c.s.r. semigroup defines walks on Euclidean
                  sphere, all its nonsingular elements are similar (in the
                  same basis) to orthogonal. We classify all nonnegative
                  c.s.r. semigroups and arbitrary low-dimensional semigroups.
                  For higher dimensions, we describe five classes and leave
                  an open problem on completeness of that list. The problem
                  of algorithmic recognition of c.s.r. property is proved to
                  be polynomially solvable for irreducible semigroups and
                  undecidable for reducible ones.",
}

@INPROCEEDINGS{Prot:CDC05-2,
 author        = "Protasov, Vladimir",
 title         = "Applications of the Joint Spectral Radius to Some Problems
                  of Functional Analysis, Probability and Combinatorics",
 booktitle     = "Proceedings of the 44th {IEEE} Conference on Decision and
                  Control and European Control Conference 2005, Seville,
                  Spain, December 12--15",
 year          = "2005",
 pages         = "3025--3030",
 doi           = "10.1109/CDC.2005.1582625",
 annote        = "In this paper we discuss applications of the joint spectral
                  characteristics of finite dimensional linear operators such
                  as joint spectral radius, lower spectral radius,
                  $p$-radius, Lyapunov exponent etc. to some problems of
                  functional analysis, fractal geometry, probability theory
                  and combinatorial number theory.",
}

@INPROCEEDINGS{Prot:CDC05-1,
 author        = "Protasov, Vladimir",
 title         = "The Geometric Approach for Computing the Joint Spectral
                  Radius",
 booktitle     = "Proceedings of the 44th {IEEE} Conference on Decision and
                  Control and European Control Conference 2005, Seville,
                  Spain, December 12--15",
 year          = "2005",
 pages         = "3001--3006",
 doi           = "10.1109/CDC.2005.1582621",
 annote        = "In this paper we describe the geometric approach for
                  computing the joint spectral radius of a finite family of
                  linear operators acting in finite-dimensional Eucledian
                  space. The main idea is to use the invariant sets of of
                  these operators. It is shown that any irreducible family of
                  operators possesses a centrally-symmetric invariant compact
                  set, not necessarily unique. The Minkowski norm generated
                  by the convex hull of an invariant set (invariant body)
                  possesses special extremal properties that can be put to
                  good use in exploring the joint spectral radius. In
                  particular, approximation of the invariant bodies by
                  polytopes gives an algorithm for computing the joint
                  spectral radius with a prescribed relative deviation
                  $\varepsilon$. This algorithm is polynomial with respect to
                  $1/\varepsilon$ if the dimension is fixed. Another
                  direction of our research is the asymptotic behavior of the
                  orbit of an arbitrary point under the action of all
                  products of given operators. We observe some relations
                  between the constants of the asymptotic estimations and the
                  sizes of the invariant bodies. In the last section we give
                  a short overview on the extension of geometric approach to
                  the $L_p$-spectral radius.",
}

@ARTICLE{Prot:SIAMJCO22,
 author        = "Protasov, Vladimir Y.",
 authauthor    = "Protasov, Vladimir",
 title         = "The {B}arabanov Norm is Generically Unique, Simple, and
                  Easily Computed",
 journal       = "SIAM J. Control Optim.",
 fjournal      = "SIAM Journal on Control and Optimization",
 year          = "2022",
 volume        = "60",
 number        = "4",
 pages         = "2246--2267",
 eprinttype    = "arXiv",
 eprint        = "2109.12159",
 issn          = "0363-0129",
 mrclass       = "93D20 (37N35 39A22 46B20 93C55)",
 mrnumber      = "4457709",
 zblnumber     = "07571322",
 doi           = "10.1137/21M1426821",
 url           = "https://epubs.siam.org/doi/10.1137/21M1426821",
 annote        = "Every irreducible discrete-time linear switching system
                  possesses an invariant convex Lyapunov function (Barabanov
                  norm), which provides a very refined analysis of
                  trajectories. Until recently that notion remained rather
                  theoretical apart from special cases. In 2015 N.Guglielmi
                  and M.Zennaro showed that many systems possess at least one
                  simple Barabanov norm, which moreover, can be efficiently
                  computed. In this paper we classify all possible Barabanov
                  norms for discrete-time systems. We prove that, under mild
                  assumptions, such norms are unique and are either
                  piecewise-linear or piecewise quadratic. Those assumptions
                  can be verified algorithmically and the numerical
                  experiments show that a vast majority of systems satisfy
                  them. For some narrow classes of systems, there are more
                  complicated Barabanov norms but they can still be
                  classified and constructed. Using those results we find all
                  trajectories of the fastest growth. They turn out to be
                  eventually periodic with special periods. Examples and
                  numerical results are presented.",
}

@ARTICLE{Prot:FU98,
 author        = "Protasov, Vladimir {\relax{}Yu}.",
 authauthor    = "Protasov, V.",
 title         = "A generalization of the joint spectral radius: the
                  geometrical approach",
 journal       = "Facta Univ. Ser. Math. Inform.",
 fjournal      = "Facta Universitatis. Series: Mathematics and Informatics",
 year          = "1998",
 number        = "13",
 pages         = "19--23",
 issn          = "0352-9665",
 mrclass       = "15A18",
 mrnumber      = "2015883 (2004g:15013)",
 mrreviewer    = "Gerard Sierksma",
 zblnumber     = "1058.15011",
 zblreviewer   = "Sini{\v{s}}a Vre{\'c}ica (Beograd)",
 annote        = "The author generalizes his earlier result on the joint
                  spectral radius of the finite set of linear operators in
                  the Euclidean space.",
}

@ARTICLE{Prot:MP15,
 author        = "Protasov, Vladimir {\relax{}Yu}.",
 authauthor    = "Protasov, Vladimir",
 title         = "Spectral simplex method",
 journal       = "Math. Program.",
 fjournal      = "Mathematical Programming. A Publication of the Mathematical
                  Optimization Society",
 year          = "2016",
 volume        = "156",
 number        = "1-2, Ser. A",
 pages         = "485--511",
 issn          = "0025-5610",
 mrclass       = "15B48 (15A42 90C26)",
 mrnumber      = "3459208",
 zblnumber     = "06562698",
 doi           = "10.1007/s10107-015-0905-2",
 url           = "https://link.springer.com/article/10.1007/s10107-015-0905-2",
 annote        = "We develop an iterative optimization method for finding the
                  maximal and minimal spectral radius of a matrix over a
                  compact set of nonnegative matrices. We consider matrix
                  sets with product structure, i.e., all rows are chosen
                  independently from given compact sets (row uncertainty
                  sets). If all the uncertainty sets are finite or
                  polyhedral, the algorithm finds the matrix with
                  maximal/minimal spectral radius within a few iterations. It
                  is proved that the algorithm avoids cycling and terminates
                  within finite time. The proofs are based on spectral
                  properties of rank-one corrections of nonnegative matrices.
                  The practical efficiency is demonstrated in numerical
                  examples and statistics in dimensions up to 500. Some
                  generalizations to non-polyhedral uncertainty sets,
                  including Euclidean balls, are derived. Finally, we
                  consider applications to spectral graph theory,
                  mathematical economics, dynamical systems, and difference
                  equations.",
}

@ARTICLE{Prot:SCL16,
 author        = "Protasov, Vladimir {\relax{}Yu}.",
 authauthor    = "Protasov, Vladimir",
 title         = "Linear switching systems with slow growth of trajectories",
 journal       = "Systems Control Lett.",
 fjournal      = "Systems \&{} Control Letters",
 volume        = "90",
 year          = "2016",
 pages         = "54--60",
 issn          = "0167-6911",
 mrclass       = "93D05 (34A38 34C11 34D20 93C30)",
 mrnumber      = "3470446",
 mrreviewer    = "Petro Feketa",
 zblnumber     = "1335.93062",
 doi           = "10.1016/j.sysconle.2016.01.006",
 url           = "https://www.sciencedirect.com/science/article/abs/pii/S0167691116000207",
 annote        = "We prove the existence of positive linear switching systems
                  (continuous time), whose trajectories grow to infinity, but
                  slower than a given increasing function. This implies that,
                  unlike the situation with linear ODE, the maximal growth of
                  trajectories of linear systems may be arbitrarily slow. For
                  systems generated by a finite set of matrices, this
                  phenomenon is proved to be impossible in dimension , while
                  in all bigger dimensions the sublinear growth may occur.
                  The corresponding examples are provided and several open
                  problems are formulated.",
}

@ARTICLE{Prot:AJIFAC19,
 author        = "Protasov, Vladimir {\relax{}Yu}.",
 authauthor    = "Protasov, Vladimir",
 title         = "Comprehensive {L}yapunov functions for linear switching
                  systems",
 journal       = "Automatica J. IFAC",
 fjournal      = "Automatica. A Journal of IFAC, the International Federation
                  of Automatic Control",
 year          = "2019",
 volume        = "109",
 pages         = "108526, 7",
 issn          = "0005-1098",
 mrclass       = "93D30 (15A06 93C30)",
 mrnumber      = "3992376",
 zblnumber     = "1429.93317",
 doi           = "10.1016/j.automatica.2019.108526",
 url           = "https://www.sciencedirect.com/science/article/pii/S0005109819303875",
 annote        = "For a linear dynamical switching system, we analyse maximal
                  asymptotic growth of trajectories depending on the initial
                  point. Both discrete and continuous time systems in
                  $\mathbb{R}^{d}$ are considered. We prove the existence of
                  a Lyapunov norm in $\mathbb{R}^{d}$ with the following
                  property: for every invariant linear subspace
                  $L\subset\mathbb{R}^{d}$ of the system, the restriction of
                  the norm on L provides a tight upper bound for the growth
                  of trajectories on $L$. For this, we introduce the concept
                  of the spectral normal form of a family of matrices.
                  Properties of the comprehensive Lyapunov norms are analysed
                  and methods of their construction are discussed.",
}

@ARTICLE{Prot:LMA21,
 author        = "Protasov, Vladimir {\relax{}Yu}.",
 authauthor    = "Vladimir Protasov",
 title         = "Antinorms on cones: duality and applications",
 journal       = "Linear Multilinear Algebra",
 year          = "2022",
 volume        = "70",
 number        = "22",
 pages         = "7387--7413",
 issn          = "0308-1087",
 mrclass       = "46B20 (37H15 52A21 93D20)",
 mrnumber      = "4571035",
 zblnumber     = "07681804",
 eprinttype    = "arXiv",
 eprint        = "2109.11882",
 doi           = "10.1080/03081087.2021.1988885",
 url           = "https://www.tandfonline.com/doi/abs/10.1080/03081087.2021.1988885",
 annote        = "An antinorm is a concave nonnegative homogeneous functional
                  on a convex cone. It is shown that if the cone is
                  polyhedral, then every antinorm has a unique continuous
                  extension from the interior of the cone. The main facts of
                  the duality theory in convex analysis, in particular, the
                  Fenchel--Moreau theorem, are generalized to antinorms.
                  However, it is shown that the duality relation for
                  antinorms is discontinuous. In every dimension there are
                  infinitely many self-dual antinorms on the positive orthant
                  and, in particular, infinitely many autopolar polyhedra.
                  For the two-dimensional case, we characterise them all. The
                  classification in higher dimensions is left as an open
                  problem. Applications to linear dynamical systems, to the
                  Lyapunov exponent of random matrix products, to the lower
                  spectral radius of nonnegative matrices, and to convex
                  trigonometry are considered.",
}

@INPROCEEDINGS{ProtJun:ECC13,
 author        = "Protasov, Vladimir {\relax{}Yu}. and Jungers, Raphael M.",
 authauthor    = "Protasov, Vladimir and Jungers, Raphael",
 title         = "Convex optimization methods for computing the {L}yapunov
                  exponent of matrices",
 booktitle     = "2013 European Control Conference (ECC)",
 year          = "2013",
 pages         = "3191--3196",
 month         = jul,
 eprinttype    = "arXiv",
 eprint        = "1201.3218",
 url           = "https://ieeexplore.ieee.org/document/6669270/",
 annote        = "We introduce a new approach to evaluate the largest Lyapunov
                  exponent of a family of nonnegative matrices. The method is
                  based on using special positive homogeneous functionals on
                  $\mathbb{R}^d_+$, which gives iterative lower and upper
                  bounds for the Lyapunov exponent. They improve previously
                  known bounds and converge to the real value. The rate of
                  converges is estimated and the efficiency of the algorithm
                  is demonstrated on several problems from applications (in
                  functional analysis, combinatorics, and language theory)
                  and on numerical examples with randomly generated matrices.
                  The method computes the Lyapunov exponent with a prescribed
                  accuracy in relatively high dimensions (up to 60). We
                  generalize this approach to all matrices, not necessarily
                  nonnegative, derive a new universal upper bound for the
                  Lyapunov exponent, and show that such a lower bound, in
                  general, does not exist.",
}

@ARTICLE{PJ:NAHS15,
 author        = "Protasov, Vladimir {\relax{}Yu}. and Jungers, Rapha{\"e}l
                  M.",
 authauthor    = "Protasov, Vladimir and Jungers, Rapha{\"e}l",
 title         = "Resonance and marginal instability of switching systems",
 journal       = "Nonlinear Anal. Hybrid Syst.",
 fjournal      = "Nonlinear Analysis. Hybrid Systems. An International
                  Multidisciplinary Journal",
 year          = "2015",
 volume        = "17",
 pages         = "81--93",
 eprinttype    = "arXiv",
 eprint        = "1411.0497",
 issn          = "1751-570X",
 mrclass       = "93D05 (37N35)",
 mrnumber      = "3351031",
 zblnumber     = "06514744",
 doi           = "10.1016/j.nahs.2015.02.003",
 url           = "https://www.sciencedirect.com/science/article/pii/S1751570X15000138",
 annote        = "We analyse the so-called Marginal Instability of linear
                  switching systems, both in continuous and discrete time.
                  This is a phenomenon of unboundedness of trajectories when
                  the Lyapunov exponent is zero. We disprove two recent
                  conjectures of \emph{Y.~Chitour, P.~Mason,} and
                  \emph{M.~Sigalotti}~\cite{CMS:SCL12} stating that for
                  generic systems, the resonance is sufficient for marginal
                  instability and for polynomial growth of the trajectories.
                  The concept of resonance originated with the same authors
                  is modified. A characterization of marginal instability
                  under some mild assumptions on the system is provided.
                  These assumptions can be verified algorithmically and are
                  believed to be generic. Finally, we analyze possible types
                  of fastest asymptotic growth of trajectories. An example of
                  a marginally unstable pair of matrices with non-polynomial
                  growth is given.",
}

@ARTICLE{PJB:SIAMJMAA10,
 author        = "Protasov, Vladimir {\relax{}Yu}. and Jungers, Rapha{\"e}l M.
                  and Blondel, Vincent D.",
 authauthor    = "Protasov, V. and Jungers, R. and Blondel, Vincent",
 title         = "Joint spectral characteristics of matrices: a conic
                  programming approach",
 journal       = "SIAM J. Matrix Anal. Appl.",
 fjournal      = "SIAM Journal on Matrix Analysis and Applications",
 year          = "2009/10",
 volume        = "31",
 number        = "4",
 pages         = "2146--2162",
 issn          = "0895-4798",
 mrclass       = "15A18 (15A60 90C22 93D20)",
 mrnumber      = "2678961 (2011g:15020)",
 mrreviewer    = "Natalia Bebiano",
 zblnumber     = "1203.65093",
 zblreviewer   = "Hans Benker (Merseburg)",
 doi           = "10.1137/090759896",
 url           = "https://epubs.siam.org/doi/10.1137/090759896",
 annote        = "The joint spectral radius $\hat{\rho}(\mathcal{M})$ of a set
                  of $n\times n$ real matrices $\mathcal{M}$ is the exponent
                  of the maximal asymptotic growth of products of matrices
                  from this set when the length of the products grows. The
                  joint spectral subradius $\breve{\rho}(\mathcal{M})$ is the
                  minimal growth counterpart, that is:
                  \[\hat{\rho}(\mathcal{M})=\lim_{k\to\infty}
                  \max\{\|A_{d_1}\cdots A_{d_k}\|^{1/k} : A_i
                  \in\mathcal{M}\},\]
                  \[\breve{\rho}(\mathcal{M})=\lim_{k\to\infty}
                  \min\{\|A_{d_1}\cdots A_{d_k}\|^{1/k} : A_i
                  \in\mathcal{M}\}.\] The authors propose a new method to
                  compute the joint spectral radius and the joint spectral
                  subradius of a set of matrices $\mathcal{M}$. The
                  efficiency of the new algorithm is demonstrated by applying
                  it to several problems in combinatorics, number theory and
                  discrete mathematics.\par The authors first restrict their
                  attention to matrices that leave a cone invariant. The
                  accuracy of their algorithm, depending on geometric
                  properties of the invariant cone, is estimated. Then they
                  extend their method to arbitrary sets of matrices by a
                  lifting procedure, and demonstrate the efficiency of the
                  new algorithm by applying it to several problems in
                  combinatorics, number theory, and discrete mathematics.",
}

@ARTICLE{PGBR:IJC11,
 author        = "Przedwojski, Marek and Galkowski, Krzysztof and Bauer, Peter
                  and Rogers, Eric",
 title         = "On the stability and control of discrete linear systems with
                  clock synchronisation errors",
 journal       = "Internat. J. Control",
 fjournal      = "International Journal of Control",
 year          = "2011",
 volume        = "84",
 number        = "9",
 pages         = "1491--1499",
 issn          = "0020-7179",
 mrclass       = "93D15 (93C30 94C10)",
 mrnumber      = "2842730",
 zblnumber     = "1230.93052",
 doi           = "10.1080/00207179.2011.604164",
 url           = "https://www.tandfonline.com/doi/abs/10.1080/00207179.2011.604164",
 annote        = "This article considers discrete linear time-invariant
                  systems that can be decomposed into subsystems whose states
                  are synchronised by a common clock with a signal that
                  reaches them with delays. In particular, stability for the
                  case where all subsystems have the same sampling frequency,
                  but different switching times, is investigated. In contrast
                  to previous work, the approach taken here models the set of
                  system matrices that arise using a polytopic uncertainty
                  description, which has seen extensive application in robust
                  control theory for linear systems. Stabilisation is then
                  achieved by state feedback and a method to handle the
                  combinatorial explosion of the number of polytope vertices
                  is developed and illustrated using an example from swarm
                  system navigation.",
}

@ARTICLE{Radj:IUMJ90,
 author        = "Radjavi, Heydar",
 title         = "On reducibility of semigroups of compact operators",
 journal       = "Indiana Univ. Math. J.",
 fjournal      = "Indiana University Mathematics Journal",
 year          = "1990",
 volume        = "39",
 number        = "2",
 pages         = "499--515",
 coden         = "IUMJAB",
 issn          = "0022-2518",
 mrclass       = "47A15 (47D03 47D25)",
 mrnumber      = "1089051 (91m:47009)",
 mrreviewer    = "{\`E}. V. Kissin",
 zblnumber     = "0691.47034",
 zblreviewer   = "H. Radjavi",
 doi           = "10.1512/iumj.1990.39.39028",
 url           = "http://www.iumj.indiana.edu/docs/39028/39028.asp",
 annote        = "A multiplicative semigroup $\mathcal{S}$ of compact
                  operators on a Hilbert space is said to be reducible if its
                  members have a common nontrivial invariant subspace.
                  Sufficient conditions for reducibility are given. One
                  sample corollary is that if 1 is the unique element of
                  largest modulus in the spectrum of every member of
                  $\mathcal{S}$, then $\mathcal{S}$ is reducible; further
                  information is obtained about the lattice of invariant
                  subspaces of $\mathcal{S}$. Another corollary concerns the
                  case in which the operators in $\mathcal{S}$ have matrices
                  with nonnegative entries: if every member of such an
                  $\mathcal{S}$ has spectral radius 1, then $\mathcal{S}$ is
                  reducible. Examples of simple irreducible semigroups are
                  also presented.",
}

@ARTICLE{RRS:PAMS00,
 author        = "Radjavi, Heydar and Rosenthal, Peter and Shulman, Victor",
 title         = "Operator semigroups with quasinilpotent commutators",
 journal       = "Proc. Amer. Math. Soc.",
 fjournal      = "Proceedings of the American Mathematical Society",
 year          = "2000",
 volume        = "128",
 number        = "8",
 pages         = "2413--2420",
 coden         = "PAMYAR",
 issn          = "0002-9939",
 mrclass       = "47A13 (47D03)",
 mrnumber      = "1706985 (2000k:47004)",
 mrreviewer    = "Cecelia Laurie",
 zblnumber     = "1099.47503",
 zblreviewer   = "F. H. Vasilescu (Villeneuve d'Ascq)",
 doi           = "10.1090/S0002-9939-00-05622-7",
 url           = "https://www.ams.org/journals/proc/2000-128-08/S0002-9939-00-05622-7/",
 annote        = "\emph{R.\,M.~Guralnick} has shown [Linear Multilinear
                  Algebra 9, 133--148 (1980)] that a multiplicative semigroup
                  of matrices $\mathcal{S}$ over any field is
                  triangularizable over an extension field if $ST-TS$ is
                  nilpotent for all $S,T\in\mathcal{S}$. Starting from this
                  result (which is re-proved in the paper for the simpler
                  case of the complex field), the authors investigate the
                  possible analogues in the infinite-dimensional case. Using
                  a recent result of
                  \emph{Yu.\,V.~Turovskii}~\cite{Turovskii:JFA99}, the
                  authors show that if $\mathcal{S}$ is a semigroup of
                  compact operators in a Hilbert space such that $ST-TS$ is
                  quasinilpotent for all $S,T\in\mathcal{S}$, then
                  $\mathcal{S}$ is triangularizable. In the case of arbitrary
                  Banach spaces, the authors show that any strongly compact
                  group of operators having quasinilpotent.",
}

@MASTERSTHESIS{Reber:PhD19,
 author        = "Reber, David Patrick",
 authauthor    = "Reber, David",
 title         = "Exponential stability of intrinsically stable dynamical
                  networks and switched networks with time-varying time
                  delays",
 school        = "Department of Mathematics",
 address       = "Brigham Young University",
 year          = "2019",
 url           = "https://scholarsarchive.byu.edu/etd/7136",
 annote        = "Dynamic processes on real-world networks are time-delayed
                  due to finite processing speeds and the need to transmit
                  data over nonzero distances. These time-delays often
                  destabilize the network's dynamics, but are difficult to
                  analyze because they increase the dimension of the
                  network.We present results outlining an alternative means
                  of analyzing these networks, by focusing analysis on the
                  Lipschitz matrix of the relatively low-dimensional
                  undelayed network. The key criteria, intrinsic stability,
                  is computationally efficient to verify by use of the power
                  method. We demonstrate applications from control theory and
                  neural networks.",
}

@ARTICLE{ReberWebb:Nonlin20,
 author        = "Reber, David and Webb, Benjamin",
 title         = "Intrinsic stability: stability of dynamical networks and
                  switched systems with any type of time-delays",
 journal       = "Nonlinearity",
 fjournal      = "Nonlinearity",
 year          = "2020",
 volume        = "33",
 number        = "6",
 pages         = "2660--2685",
 issn          = "0951-7715,1361-6544",
 mrclass       = "37C75 (15A42 37H05)",
 mrnumber      = "4105373",
 mrreviewer    = "Vahid S. Bokharaie",
 zblnumber     = "1443.37029",
 doi           = "10.1088/1361-6544/ab7728",
 url           = "https://iopscience.iop.org/article/10.1088/1361-6544/ab7728",
 annote        = "In real-world networks, interactions between network
                  elements are inherently time-delayed. These time-delays can
                  both slow and destabilize the network, leading to poor
                  performance. However, not all networks can be destabilized
                  by time-delays. Previously, it has been shown that if a
                  network is intrinsically stable, it maintains stability
                  when constant time-delays are introduced. Here we show that
                  intrinsically stable networks and a broad class of switched
                  systems remain stable in the presence of any type of
                  time-varying time-delays whether these delays are periodic,
                  stochastic, or otherwise. We apply these results to a
                  number of well-studied systems to demonstrate that
                  intrinsic stability is both computationally inexpensive and
                  improves on previous stability methods. Furthermore, we
                  prove that the asymptotic state of an intrinsically stable
                  switched system is independent of both the system's initial
                  conditions and the presence of time-varying time-delays.
                  Thus, we justify ignoring the occurance of any type of
                  time-delays when modeling intrinsically stable real-world
                  systems for asymptotic analysis, and provide an efficient
                  means of engineering delay-robust systems.",
}

@ARTICLE{Robert:LAA69,
 author        = "Robert, F.",
 title         = "Blocs-{$H$}-matrices et convergence des m{\'e}thodes
                  it{\'e}ratives classiques par blocs",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "1969",
 volume        = "2",
 pages         = "223--265",
 mrclass       = "65.35",
 mrnumber      = "0250463 (40 \#3700)",
 mrreviewer    = "B. N. Parlett",
 zblnumber     = "0182.21302",
 doi           = "10.1016/0024-3795(69)90029-9",
 url           = "https://www.sciencedirect.com/science/article/pii/0024379569900299",
 language      = "english",
 annote        = "This long paper is based on the author's Ph. D. thesis at
                  the University of Grenoble. Utilizing the notion of a
                  regular vectorial norm $p$ and the associated majorants of
                  a given matrix $A$, the author calls $A$ a block-$H$-matrix
                  relative to $p$ [cf. \emph{A.~Ostrovski}, J. math. Analysis
                  Appl. 2, 351--369 (1961)] if the corresponding block Jacobi
                  matrix has a convergent majorant. It is proven that if $A$
                  is a block-$H$-matrix, then the corresponding block
                  Gauss--Seidel and SQR matrices are convergent. Schroeder's
                  algorithm for bounding the error of a linear Iterative
                  method [MRC Technol. Rep. 242, Wisconsin, July 1961] is
                  generalized to the case of blocks and it is shown how to
                  utilize the algorithm in conjunction with the block
                  Gauss--Seldel and SQR methods.",
}

@ARTICLE{Robert70,
 author        = "Robert, Fran{\c{c}}ois",
 title         = "M{\'e}thodes it{\'e}ratives ``s{\'e}rie parall{\`e}le''",
 journal       = "C. R. Acad. Sci. Paris S{\'e}r. A-B",
 fjournal      = "Comptes Rendus Hebdomadaires des S{\'e}ances de
                  l'Acad{\'e}mie des Sciences. S{\'e}ries A et B",
 year          = "1970",
 volume        = "271",
 pages         = "A847--A850",
 mrclass       = "65.10",
 mrnumber      = "0269076 (42 \#3972)",
 zblnumber     = "0209.17602",
 annote        = "Il s'agit dans cette note, de pr{\'e}senter une classe
                  d'algorithmes d'it{\'e}rations lin{\'e}aires, pour la
                  r{\'e}solution d'un syst{\`e}me lin{\'e}aire de $n = q\cdot
                  k$ {\'e}quations, adapt{\'e}s {\`a} un calcul en
                  parall{\`e}le mettant en oeuvre $q$ ordinateurs. On appelle
                  proc{\'e}d{\'e} s{\'e}rie parall{\`e}le $\{k,q\}$ un tel
                  proc{\'e}d{\'e}, qui est finalement le suivant: les $q$
                  ordinateurs traitent simultan{\'e}ment les $q$
                  premi{\`e}res composantes (chacun en prend une en charge)
                  selon la m{\'e}thode de Jacobi puis mettent en commun leurs
                  r{\'e}sultats, puis traitent ensuite en parall{\`e}le les
                  $q$ composantes suivantes etc. Il est clair que le
                  proc{\'e}d{\'e} $\{1,n\}$ est alors celui de Jacobi
                  habituel, tandis que le proc{\'e}d{\'e} $\{n,1\}$ est celui
                  de Gauss--Seidel. L'{\'e}tude de la contraction (qui
                  implique la convergence) relativement {\`a} une norme
                  vectorielle permet de donner une condition (utilisable
                  num{\'e}riquement) de convergence, et d'obtenir les
                  r{\'e}sultats suivants: l) Pour une $M$-matrice, tous les
                  proc{\'e}d{\'e}s s{\'e}rie parall{\`e}le convergent. 2) A
                  fortiori, pour une $M$-matrice, tous les proc{\'e}d{\'e}s
                  s{\'e}rie-parall{\`e}le sont convergents. Si, de plus,
                  k'est multiple de $k$, le proc{\'e}d{\'e} $\{k',q'\}$ (avec
                  {\'e}videmment $k'q' =n = kq$) converge plus vite que le
                  proc{\'e}d{\'e} $\{k,q\}$. On compl{\`e}te donc ainsi le
                  r{\'e}sultat classique de comparaison de Jacobi et
                  Gauss--Seidel, qui est cas particulier pour $k' =n$ et $k=
                  1$. Il en r{\'e}sulte que pour une $M$-matrice, les
                  proc{\'e}d{\'e}s ainsi d{\'e}finis convergent au moins
                  aussi vite que celui de Jacobi, mais ne convergent pas plus
                  vite que celui de Gauss--Seidel.",
}

@ARTICLE{Robert77,
 author        = "Robert, Fran{\c{c}}ois",
 title         = "Convergence locale d'it{\'e}rations chaotiques non
                  lin{\'e}aires",
 journal       = "C. R. Acad. Sci. Paris S{\'e}r. A-B",
 fjournal      = "Comptes Rendus Hebdomadaires des S{\'e}ances de
                  l'Acad{\'e}mie des Sciences. S{\'e}ries A et B",
 year          = "1977",
 volume        = "284",
 number        = "12",
 pages         = "A679--A682",
 issn          = "0151-0509",
 mrclass       = "65J05",
 mrnumber      = "431670",
 zblnumber     = "0363.65053",
 annote        = "Sufficient conditions are given to ensure local convergence
                  of chaotic iterations for non-linear fixed point equations
                  in several variables. This result, the proof of which is
                  rather technical, extends a classical result from Ostrowski
                  (local convergence of successive approximations) to chaotic
                  iterations.",
}

@ARTICLE{RSS:LAA10,
 author        = "Rodman, Leiba and Seyalioglu, Hakan and Spitkovsky, Ilya
                  M.",
 authauthor    = "Rodman, Leiba and Seyalioglu, Hakan and Spitkovsky, Ilya",
 title         = "On common invariant cones for families of matrices",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2010",
 volume        = "432",
 number        = "4",
 pages         = "911--926",
 eprinttype    = "arXiv",
 eprint        = "0903.0444",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15B48 (05A05)",
 mrnumber      = "2577636 (2011d:15051)",
 mrreviewer    = "Maria Chiara Brambilla",
 zblnumber     = "1188.15028",
 zblreviewer   = "Juan Ramon Torregrosa Sanchez (Valencia)",
 doi           = "10.1016/j.laa.2009.10.004",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379509005187",
 annote        = "The authors consider the existence and construction of
                  common invariant cones for families of real matrices.\par
                  Let $\mathcal{A}={A_1,A_2,\ldots,A_n}$ be a family of
                  $2\times 2$ real matrices. The authors obtain necessary and
                  necessary and sufficient conditions for the existence of an
                  $\mathcal{A}$-invariant proper cone. They analyze different
                  cases:\par (i) when all the matrices $A_j$ share a dominant
                  eigenvector $u$;\par (ii) when all the elements of
                  $\mathcal{A}_1$ are diagonalizable matrices, being
                  \[\mathcal{A}_1 =\mathcal{A}\cup\{A_i A_j : A_i, A_j
                  \in\mathcal{A},\quad \det A_i <0, \det A_j
                  <0~\text{and}~A_i A_j \neq cI\};\]\indent (iii) when there
                  is none, one or two dominant eigenlines (one dimensional
                  eigenspace) corresponding to non-diagonalizable matrices in
                  $\mathcal{A}_1$;\ldots.\par For matrices of arbitrary size
                  $m\times m$, the authors give an existence criterion for
                  (and actually a construction of) a common invariant cone of
                  a family $\mathcal{A}={A_1,A_2,\ldots,A_n}$ under the
                  assumption that all the elements A i can be put in a
                  diagonal form by the same similarity transformation. They
                  also provide some sufficient conditions for such a cone to
                  exist when the matrices share the dominant eigenvector.\par
                  Finally, the authors present several examples illustrating
                  the theoretical results and their limitations.",
}

@ARTICLE{RS:PAMS95,
 author        = "Rosenthal, P. and So{\l}tysiak, A.",
 title         = "Formulas for the joint spectral radius of non-commuting
                  {B}anach algebra elements",
 journal       = "Proc. Amer. Math. Soc.",
 fjournal      = "Proceedings of the American Mathematical Society",
 year          = "1995",
 volume        = "123",
 number        = "9",
 pages         = "2705--2708",
 coden         = "PAMYAR",
 issn          = "0002-9939",
 mrclass       = "47A13 (46H05)",
 mrnumber      = "1257123 (95k:47008)",
 mrreviewer    = "F.-H. Vasilescu",
 zblnumber     = "0849.46034",
 doi           = "10.2307/2160564",
 url           = "https://www.jstor.org/stable/2160564",
 annote        = "In 1960, \emph{G.-C.~Rota} and
                  \emph{G.~Strang}~\cite{RotaStr:IM60} proposed a formula for
                  the joint spectral radius of an $n$-tuple (or, more
                  generally, any bounded set) of Banach algebra elements. A
                  different formula was investigated by \emph{M.\,A.~Berger}
                  and \emph{Y.~Wang}~\cite{BerWang:LAA92}. We consider
                  relations between these formulas and the ``geometric
                  spectral radius''.",
}

@BOOK{Rota03,
 author        = "Rota, Gian-Carlo",
 title         = "Gian-{C}arlo {R}ota on analysis and probability",
 series        = "Contemporary Mathematicians",
 publisher     = "Birkh{\"a}user Boston Inc.",
 address       = "Boston, MA",
 year          = "2003",
 pages         = "xxx+381",
 isbn          = "0-8176-4275-7",
 mrclass       = "01A75 (28-03 47-06 60-03 92D20)",
 mrnumber      = "1944526 (2004c:01047)",
 mrreviewer    = "Irene Hueter",
 zblnumber     = "1159.01014",
 note          = "Selected papers and commentaries, Edited by Jean Dhombres,
                  Joseph P. S. Kung and Norton Starr",
 annote        = "This volume consists of selected papers by Gian-Carlo Rota,
                  a rare mathematician who made major contributions to
                  several areas of mathematics. Presented in Part I are his
                  papers in analysis, written at the beginning of his career
                  and having a continuing and pervasive influence today. Part
                  II is devoted to his papers on convexity and probability
                  theory, written towards the end of his career and
                  containing many ideas that have yet to be fully developed.
                  Comprehensive commentaries by experts in the field are
                  included in every chapter. Accessible to both experts and
                  beginners. Gian-Carlo Rota was one of those rare
                  mathematicians who made major contributions to several
                  areas of mathematics. Presented in the first part of this
                  volume are reprints of his papers in analysis, which were
                  written at the beginning of his career. These papers on
                  differential equations, operator theory, ergodic theory,
                  and other subjects have a continuing and pervasive
                  influence. Reprints of his papers on convexity and
                  probability theory are presented in the second part of the
                  work. These were written towards the end of his career and
                  contain many ideas that have yet to be fully developed.
                  Comprehensive commentaries are included in every chapter.
                  These survey articles detail work inspired by Rota's papers
                  and also include discussions of many unsolved problems. As
                  is customary with Rota's writings, the papers included in
                  the volume -- some published here for the first time --
                  contain many fresh and unexpected ideas for further
                  research. Thus, this volume will be of interest to both
                  experts and beginners in the above-mentioned fields.",
}

@ARTICLE{RotaStr:IM60,
 author        = "Rota, Gian-Carlo and Strang, Gilbert",
 title         = "A note on the joint spectral radius",
 journal       = "Koninklijke Nederlandse Akademie van Wetenschappen.
                  Indagationes Mathematicae",
 year          = "1960",
 volume        = "22",
 pages         = "379--381",
 mrclass       = "47.10",
 mrnumber      = "0147922 (26 \#5434)",
 mrreviewer    = "H. Heuser",
 zblnumber     = "0095.09701",
 doi           = "10.1016/S1385-7258(60)50046-1",
 url           = "https://www.sciencedirect.com/science/article/pii/S1385725860500461",
 annote        = "F{\"u}r eine beschr{\"a}nkte Untermenge $B$ einer normierten
                  Algebra $\mathfrak{A}$ mit Einheitselement wird der
                  vereinigte (joint) Spelctralradius durch $r(B)=
                  \lim_{n\to\infty} \sup_{t\in P_n} \|T\|^{1/n}$ definiert;
                  hierin ist $P_n$ die Menge aller Produkte mit $n$ Faktoren
                  aus $B$. Ist $\mathfrak{R}$ die Menge der (multiplikativen)
                  Normen von $\mathfrak{A}$, die zu der gegebenen Norm von
                  $\mathfrak{A}$ {\"a}quivalent sind (in dem Sinne, da{\ss}
                  $k \|T\| \le N(T)\le k^{-1}\|T\|$ f{\"u}r eine Konstante
                  $k\neq 0$ und alle $T$ in $\mathfrak{A}$, so ist $r(B) =
                  \inf_{N\in\mathfrak{R}} \sup_{t\in B} N(T)$. Dieser Satz
                  wird mit Hilfe eines Lemmas bewiesen, aus dem die Autoren
                  {\"u}berdies noch eine Bemerkung {\"u}ber die
                  Fortsetzbarkeit einer auf der Unteralgebra
                  $\mathfrak{A}'\subset\mathfrak{A}$ definierton Norm
                  gewinnen.",
}

@ARTICLE{RotaStr:IMP60,
 author        = "Rota, Gian-Carlo and Strang, Gilbert",
 title         = "A note on the joint spectral radius",
 journal       = "Koninklijke Nederlandse Akademie van Wetenschappen.
                  Indagationes Mathematicae (Proceedings)",
 year          = "1960",
 volume        = "63",
 pages         = "379--381",
 mrclass       = "47.10",
 mrnumber      = "0147922 (26 \#5434)",
 mrreviewer    = "H. Heuser",
 zblnumber     = "0095.09701",
 doi           = "10.1016/S1385-7258(60)50046-1",
 url           = "https://www.sciencedirect.com/science/article/pii/S1385725860500461",
 annote        = "The notion of joint spectral radius of a set of elements of
                  a normed algebra, was obtained in the course of some work
                  in matrix theory. It was later noticed that the same
                  considerations are valid in any normed algebra,
                  irrespective of dimension. The notion seems to be useful
                  enough in certain contexts to warrant the following
                  elementary discussion.\par Let $\mathbf{B}$ be any bounded
                  subset of the normed algebra $\mathfrak{A}$ with identity
                  $e$. Let $P_{n}$ be the set of all elements of
                  $\mathfrak{A}$ which are the products of $n$ elements of
                  $\mathbf{B}$. The joint spectral radius of the set
                  $\mathbf{B}$ is defined to be the nonnegative number
                  $r(\mathbf{B})=\lim_{n\to\infty} \sup_{T\in
                  P_{n}}\|T\|^{1/n}$. This number is well-defined.
                  Intuitively, the joint spectral radius is related to the
                  commutativity of the elements of $\mathbf{B}$. We prove a
                  result connecting the joint spectral radius with the
                  equivalent norms the algebra may be provided with. Let
                  $\mathfrak{N}$ be the family of norms on $\mathfrak{A}$
                  under which $\mathfrak{A}$ is a normed algebra, equivalent
                  to the given norm $N(T)$ on $\mathfrak{A}$ in the sense
                  that $k\|T\|<N(T)<k^{-1}\|T\|$ for some constant $k \neq 0$
                  and for all $T$ in $\mathfrak{A}$. Note that all the norms
                  in $\mathfrak{N}$ are assumed to be multiplicative.\par
                  \textbf{Proposition 1}.
                  $r(\mathbf{B})=\inf_{N\in\mathfrak{N}}\sup_{\mathbf{T\in
                  B}}N(T)$.\par In other words, the joint spectral radius of
                  a bounded set $\mathbf{B}$ is equal to the infimum of the
                  ``norm'' of the set, ranging over all equivalent norms.
                  This result will be a consequence of the following\par
                  \textbf{Lemma}. \emph{Let $\mathbf{S}$ be any subset of
                  $\mathfrak{A}$. A necessary and sufficient condition that
                  there exist an equivalent norm $N$ such that $N(T)<1$ for
                  all $T\in\mathbf{S}$ is that there exist a constant $K$
                  such that $\left\|T_{1} T_{2} \ldots T_{n}\right\|<K$ for
                  any $T_{1}, T_{2}, \ldots, T_{n}$ in $\mathbf{S}$ and for
                  all $n$.}",
}

@ARTICLE{SafKhan:IEEETAC17,
 author        = "Safavi, Sam and Khan, Usman A.",
 authauthor    = "Safavi, Sam and Khan, Usman",
 title         = "Asymptotic stability of {LTV} systems with applications to
                  distributed dynamic fusion",
 journal       = "IEEE Trans. Automat. Control",
 year          = "2017",
 volume        = "62",
 number        = "11",
 pages         = "5888--5893",
 eprinttype    = "arXiv",
 eprint        = "1412.8018",
 issn          = "0018-9286",
 mrclass       = "93C55 (39A30 93D20)",
 mrnumber      = "3730968",
 zblnumber     = "1390.93506",
 doi           = "10.1109/TAC.2017.2648501",
 url           = "https://ieeexplore.ieee.org/document/7807326",
 annote        = "We investigate the behavior of Linear Time-Varying (LTV)
                  systems with randomly appearing, sub-stochastic system
                  matrices. Motivated by dynamic fusion over mobile networks,
                  we develop conditions on the system matrices that lead to
                  asymptotic stability of the underlying LTV system. By
                  partitioning the sequence of system matrices into slices,
                  we obtain the stability conditions in terms of slice
                  lengths and introduce the notion of unbounded connectivity,
                  i.e., the time-intervals, over which the multi-agent
                  network is connected, do not have to be bounded as long as
                  they do not grow faster than a certain exponential rate. We
                  further apply the above analysis to derive the asymptotic
                  behavior of a dynamic leader-follower algorithm.",
}

@TECHREPORT{Saouter:inria96,
 author        = "Saouter, Yannick",
 title         = "The Mortality of a Pair of 2x2 Matrices is Decidable",
 institution   = "INRIA",
 affiliation   = "API - INRIA - IRISA",
 year          = "1996",
 number        = "RR-2842",
 type          = "Rapport de recherche",
 url           = "https://hal.inria.fr/inria-00073848",
 hal_id        = "inria-00073848",
 hal_version   = "v1",
 annote        = "A pair of matrices is said to be mortal if there is a serie
                  of these matrices for which the product is the null matrix.
                  A recent result have established that the general problem
                  of the mortality of a pair of integral matrices is
                  undecidable. In this article, we prove by using only linear
                  algebra that the mortality of a pair of $2\times 2$
                  integral matrices is decidable.",
}

@BOOK{Seneta06,
 author        = "Seneta, E.",
 title         = "Non-negative matrices and {M}arkov chains",
 series        = "Springer Series in Statistics",
 publisher     = "Springer",
 address       = "New York",
 year          = "2006",
 pages         = "xvi+287",
 isbn          = "978-0387-29765-1; 0-387-29765-0",
 mrclass       = "60-02 (15A48 60J10)",
 mrnumber      = "2209438",
 zblnumber     = "1099.60004",
 note          = "Revised reprint of the second (1981) edition
                  [Springer-Verlag, New York; MR0719544]",
 annote        = "This book is a photographic reproduction of the book of the
                  same title published in 1981, for which there has been
                  continuing demand on account of its accessible technical
                  level. Its appearance also helped generate considerable
                  subsequent work on inhomogeneous products of matrices. This
                  printing adds an additional bibliography on coefficients of
                  ergodicity and a list of corrigenda.",
}

@PHDTHESIS{Sert:Thesis16,
 author        = "Sert, Cagri",
 title         = "Joint Spectrum and Large Deviation Principles for Random
                  Products of Matrices",
 school        = "Universit{\'e} Paris-Saclay",
 year          = "2016",
 number        = "2016SACLS500",
 type          = "Theses",
 month         = Dec,
 hal_id        = "tel-01654424",
 hal_version   = "v1",
 url           = "https://theses.hal.science/tel-01654424",
 annote        = "After giving a detailed introduction and the presentation of
                  an explicit example to illustrate our study in Chapter 1,
                  we exhibit some general tool sand techniques in Chapter 2.
                  Subsequently, In Chapter 3, we prove the existence of a
                  large deviation principle (LDP) with a convex rate
                  function, forthe Cartan components of the random walks on
                  linear semisimple groups $G$. The main hypothesis is on the
                  support $S$ of the probability measure in question, and
                  asks $S$ to generate a Zariski dense semigroup. In Chapter
                  4, we introduce a limit object (a subset of the Weyl
                  chamber) that we associate to a boundedsubset $S$ of $G$.
                  We call this the joint spectrum $J(S)$ of $S$. We study its
                  properties and show that for a subset $S$ generating a
                  Zariski dense semigroup, $J(S)$ is convex body, i.e. a
                  convex compact subset of nonempty interior. We relate the
                  joint spectrum with the classical notion of joint spectral
                  radius and the rate function of LDP for random walks on
                  $G$. In Chapter 5, we introduce an exponential counting
                  function for a nite $S$ in $G$. We study its properties,
                  relate it to joint spectrum of $S$ and prove a dense
                  exponential growth theorem. In Chapter 6, we prove the
                  existence of an LDPfor Iwasawa components of random walks
                  on $G$. The hypothesis asks for a condition of absolute
                  continuity of the probability measure with respect to the
                  Haar measure. In Chapter 7, we develop some tools to tackle
                  aquestion of Breuillard on the rigidity of spectral
                  radiusof a random walk on a free group. We prove a weaker
                  geometric rigidity result.",
}

@ARTICLE{Sert:CRMASP17,
 author        = "Sert, Cagri",
 title         = "Joint spectrum and large deviation principle for random
                  matrix products",
 journal       = "C. R. Math. Acad. Sci. Paris",
 fjournal      = "Comptes Rendus Math{\'e}matique. Acad{\'e}mie des Sciences.
                  Paris",
 year          = "2017",
 volume        = "355",
 number        = "6",
 pages         = "718--722",
 issn          = "1631-073X",
 mrclass       = "60B20 (60F10)",
 mrnumber      = "3661555",
 zblnumber     = "1373.60017",
 doi           = "10.1016/j.crma.2017.04.015",
 url           = "https://www.sciencedirect.com/science/article/pii/S1631073X17301292",
 annote        = "The aim of this note is to announce some results about the
                  probabilistic and deterministic asymptotic properties of
                  linear groups. The first one is the analogue, for norms of
                  random matrix products, of the classical theorem of
                  Cram{\'e}r on large deviation principles (LDP) for sums of
                  iid real random variables. In the second result, we
                  introduce a limit set describing the asymptotic shape of
                  the powers $S^{n}=\{g_{1},\ldots,g_{n}\,|\, g_{i}\in
                  S\}$ of a subset $S$ of a semisimple linear Lie group $G$
                  (e.g. SL$(d,\mathbb{R})$). This limit set has applications,
                  among others, in the study of large deviations.",
}

@ARTICLE{Shang:FM17,
 author        = "Shang, Yilun",
 title         = "Subspace confinement for switched linear systems",
 journal       = "Forum Math.",
 fjournal      = "Forum Mathematicum",
 year          = "2017",
 volume        = "29",
 number        = "3",
 pages         = "693--699",
 issn          = "0933-7741",
 mrclass       = "15A60 (37C70 39A05 40A20 47A30)",
 mrnumber      = "3641673",
 zblnumber     = "1360.15003",
 doi           = "10.1515/forum-2015-0188",
 url           = "https://www.degruyter.com/view/journals/form/29/3/article-p693.xml",
 annote        = "In this note, we introduce the asymptotic subspace
                  confinement problem, generalizing the usual concept of
                  convergence in discrete-time linear systems. Instead of
                  precise convergence, subspace confinement only requires the
                  convergence of states to a certain subspace of the state
                  space, offering useful flexibility and applicability. We
                  establish a criterion for deciding the asymptotic subspace
                  confinement, drawing upon a general finiteness result for
                  the infinite product of matrices. Our results indicate that
                  the asymptotic subspace confinement problem is
                  algorithmically decidable when an invariant subspace for
                  the set of matrices and some polytope norms are given.",
}

@ARTICLE{Sheipak:MZ98,
 author        = "She{\u\i}pak, I. A.",
 authauthor    = "She{\u\i}pak, I.",
 title         = "Nontrivial fractals in the plane, and linear operators with
                  a joint spectral radius equal to {$1$}",
 journal       = "Math. Notes",
 fjournal      = "Mathematical Notes",
 year          = "1998",
 volume        = "63",
 number        = "5",
 pages         = "701--705",
 issn          = "0001-4346",
 mrclass       = "15A18 (28A80 47A13)",
 mrnumber      = "1683656 (2000a:15016)",
 mrreviewer    = "J. S. Joel",
 zblnumber     = "0938.37028",
 zblreviewer   = "Messoud Efendiev (Berlin)",
 doi           = "10.1007/BF02312855",
 url           = "https://link.springer.com/article/10.1007/BF02312855",
 annote        = "Properties of invariant bodies of a family of linear
                  operators $A_1 ,\ldots,A_k :\mathbb{R}^n\to\mathbb{R}^n$
                  have been studied in the paper of
                  \emph{V.\,Yu.~Protasov}~\cite{Prot:FPM96:e}, which also
                  contains an algorithm that calculates the joint spectral
                  radius of these operators. In the examples considered in
                  the paper of Protasov (loc. cit), the dimensions (both
                  topological and Hausdorff) of the invariant sets on the
                  plane are equal to either 0 or 1. The main goal of this
                  paper is to construct a class of operators with invariant
                  sets having fractional Hausdorff dimension.",
}

@ARTICLE{Shen:LAA00,
 author        = "Shen, Jianhong",
 title         = "Compactification of a set of matrices with convergent
                  infinite products",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2000",
 volume        = "311",
 number        = "1-3",
 pages         = "177--186",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15A60 (15A18 40A20)",
 mrnumber      = "1758212 (2001c:15039)",
 mrreviewer    = "K. Chandrasekhara Rao",
 zblnumber     = "0958.15021",
 zblreviewer   = "T.Nono (Hiroshima)",
 doi           = "10.1016/S0024-3795(00)00080-X",
 url           = "https://www.sciencedirect.com/science/article/pii/S002437950000080X",
 annote        = "The author generalizes and unifies some known results on a
                  set $\Sigma$ of square matrices (real or complex) with
                  right-convergent-products (RCPs). Here $\Sigma$ is said to
                  be an RCP set if for any sequence
                  $(A_1,A_2,\ldots)\in\Sigma^{\infty}$, $\lim_{n\to\infty}
                  A_1 A_2\cdots A_n$ exists. A K{\"o}nig chain of $\Sigma$
                  with respect to a given norm $\|\cdot\|$ is a sequence
                  $(A_1,A_2,\ldots)\in\Sigma^{\infty}$ such that $\|A_1
                  A_2\cdots A_m\|^{1/m}\ge \hat{\rho}$, $m=1,2,\ldots$.\par
                  In Section 2 the author studies some important properties
                  that are shared by both a bounded matrix set and its
                  closure. In Section 3 he proves the existence of a
                  K{\"o}nig chain in a compact set of matrices, and then
                  shows applications by reproducing some major known
                  results.",
}

@ARTICLE{ShenHu:SIAMJCO12,
 author        = "Shen, Jinglai and Hu, Jianghai",
 title         = "Stability of discrete-time switched homogeneous systems on
                  cones and conewise homogeneous inclusions",
 journal       = "SIAM J. Control Optim.",
 fjournal      = "SIAM Journal on Control and Optimization",
 year          = "2012",
 volume        = "50",
 number        = "4",
 pages         = "2216--2253",
 coden         = "SJCODC",
 issn          = "0363-0129",
 mrclass       = "93Dxx (34A60 93C30 93C55)",
 mrnumber      = "2974737",
 zblnumber     = "1252.93068",
 doi           = "10.1137/110845215",
 url           = "https://epubs.siam.org/doi/10.1137/110845215",
 annote        = "This paper presents a stability analysis of switched
                  homogeneous systems on cones under arbitrary and optimal
                  switching rules with extensions to conewise homogeneous or
                  linear inclusions. Several interrelated approaches, such as
                  the joint spectral radius approach and the generating
                  function approach, are exploited to derive necessary and
                  sufficient stability conditions and to develop suitable
                  algorithms for stability tests. Specifically, the
                  generalized joint spectral radius and the generalized joint
                  lower spectral radius are introduced to characterize the
                  radii of domains of strong and weak attraction.
                  Furthermore, strong and weak generating functions and their
                  radii of convergence are employed to derive stability
                  conditions; their analytic properties, numerical
                  approximations, and convergence analysis are established.
                  Extensions to conewise homogeneous or linear inclusions are
                  made to address state-dependent switching dynamics.
                  Relations between different stability notions in the strong
                  or weak sense are studied; Lyapunov techniques are used for
                  stability analysis of the conewise linear inclusions.",
}

@ARTICLE{Shih:TM99,
 author        = "Shih, Mau-Hsiang",
 title         = "{F}ritz {J}ohn's Convexity Theorem and Discrete Dynamics",
 journal       = "Trends Math.",
 fjournal      = "Trends in Mathematics. Information Center for Mathematical
                  Sciences",
 year          = "1999",
 volume        = "2",
 number        = "1",
 pages         = "66--68",
 url           = "https://www.yumpu.com/en/document/read/5014803",
 annote        = "We consider the result of great impact by Fritz John in his
                  work on Convex Geometry. John showed in 1948 that the
                  boundary of any convex region centrally symmetric with
                  respect to a point in $\mathbf{R}^n$ lies between two
                  concentric homothetic ellipsoids of ratio $1/\sqrt{n}$.
                  This result has become very important for Geometric
                  Algorithms. The purpose of this talk is to discuss its
                  matrix-theoretic version and its new application to
                  Discrete Dynamics. The content is mainly taken from papers:
                  \emph{T.~Ando} and
                  \emph{M.-H.~Shih}~\cite{AndoShih:SIAM:98},
                  \emph{M.-H.~Shih}~\cite{Shih:LAA99}.",
}

@ARTICLE{Shih:LAA99,
 author        = "Shih, Mau-Hsiang",
 title         = "Simultaneous {S}chur stability",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "1999",
 volume        = "287",
 number        = "1-3",
 pages         = "323--336",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15A18 (15A60 34D20 34K20)",
 mrnumber      = "1662876 (99k:15019)",
 mrreviewer    = "Mihail Voicu",
 zblnumber     = "0948.15016",
 zblreviewer   = "Valeriu Prepeli{\c{t}}\u{a} (Bucure{\c{s}}ti)",
 doi           = "10.1016/S0024-3795(98)10071-X",
 url           = "https://www.sciencedirect.com/science/article/pii/S002437959810071X",
 note          = "Special issue celebrating the 60th birthday of Ludwig
                  Elsner",
 annote        = "The notion of simultaneous Schur stability for a set of
                  $n\times n$ complex matrices is introduced. For a set
                  $\Sigma\subset \mathbb{C}^{n\times n}$,
                  $\mathcal{S}(\Sigma)$ denotes the multiplicative semigroup
                  generated by $\Sigma$. For an $n\times n$ complex matrix
                  $A$, $\sigma(A)$ and $r(A)$ denote the spectrum of $A$ and
                  the spectral radius of $A$, respectively. A set
                  $\Sigma\subset\mathbb{C}^{n\times n}$ is said to be
                  simultaneously Schur stable if $r(A)\le 1$, for every $A\in
                  \mathcal{S}(\Sigma)$ and $1\notin\sigma(A)$, for every
                  $A\in\overline{\mathcal{S}(\Sigma)}$. A set
                  $\Sigma\subset\mathbb{C}^{n\times n}$ is said to be
                  asymptotically stable if there is a norm $\|\cdot\|$ on
                  $\mathbb{C}^n$ and $\alpha>0$ such that $\|A\|\le\alpha<1$,
                  for every $A\in \Sigma$. The main theorem states:\par
                  \textbf{Theorem.} \emph{For a bounded set $\Sigma$ of
                  $n\times n$ complex matrices, $\Sigma$ is asymptotically
                  stable iff $\Sigma$ is simultaneously Schur stable.}\par
                  This result is used to give an analytic-combinatorial proof
                  of the generalized Gelfand spectral radius formula.\par
                  Another interesting application of the main theorem is a
                  simplified proof of a result concerning infinite products
                  of matrices. It is emphasized that in a compact
                  multiplicative semigroup of $n\times n$ complex matrices,
                  the essential nature of simultaneous Schur stability is
                  that zero is the only projection in the semigroup.\par
                  Finally, three conjectures are derived from the finiteness
                  conjecture for the generalized spectral radius.",
}

@ARTICLE{Shih:LAA01,
 author        = "Shih, Mau-Hsiang",
 title         = "K{\"o}nig chain for compact matrix sets",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2001",
 volume        = "330",
 number        = "1-3",
 pages         = "205--208",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15A60",
 mrnumber      = "1826655 (2002a:15037)",
 zblnumber     = "0982.15035",
 doi           = "10.1016/S0024-3795(01)00265-8",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379501002658",
 annote        = "The author considers a compact set $\Sigma$ of complex
                  $(n\times n)$-matrices. It is shown that $\Sigma$ admits a
                  so-called K{\"o}nig chain, i.e., a sequence $\{A_k\}$ in
                  $\Sigma$ such that $\|A_1 A_2 \cdots A_k \|^{1/m}$ is
                  greater or equal to the joint spectral radius of $\Sigma$
                  for all operator norms $\|\cdot\|$ (cf. \emph{G.-C.~Rota}
                  and \emph{G.~Strang}~\cite{RotaStr:IM60}).",
}

@ARTICLE{ShihPang:AJIFAC08,
 author        = "Shih, Mau-Hsiang and Pang, Chin-Tzong",
 title         = "Simultaneous {S}chur stability of interval matrices",
 journal       = "Automatica J. IFAC",
 fjournal      = "Automatica. A Journal of IFAC, the International Federation
                  of Automatic Control",
 year          = "2008",
 volume        = "44",
 number        = "10",
 pages         = "2621--2627",
 coden         = "ATCAA9",
 issn          = "0005-1098",
 mrclass       = "93D09",
 mrnumber      = "2531059 (2010d:93125)",
 zblnumber     = "1155.93416",
 doi           = "10.1016/j.automatica.2008.02.026",
 url           = "https://www.sciencedirect.com/science/article/pii/S0005109808002239",
 annote        = "Interval matrix structures are ubiquitous in nature and
                  engineering. Ordinarily, in an uncertain system there is
                  associated with a set of coupled interval matrices, a basic
                  issue of exploring its asymptotic stability. Here we
                  introduce the notion of simultaneous Schur stability by
                  linking the concepts of the majorant and the joint spectral
                  radius, and prove the asymptotic stability of a set of
                  interval matrices governed by simultaneous Schur stability.
                  The present result may lead to the stability analysis of
                  discrete dynamical interval systems.",
}

@ARTICLE{ShihWP:LAA97,
 author        = "Shih, Mau-Hsiang and Wu, Jinn-Wen and Pang, Chin-Tzong",
 title         = "Asymptotic stability and generalized {G}elfand spectral
                  radius formula",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "1997",
 volume        = "252",
 pages         = "61--70",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15A18",
 mrnumber      = "1428628 (97k:15028)",
 zblnumber     = "0873.15012",
 zblreviewer   = "M. M. Konstantinov (Sofia)",
 doi           = "10.1016/0024-3795(95)00592-7",
 url           = "https://www.sciencedirect.com/science/article/pii/0024379595005927",
 annote        = "Let $\Sigma\subset \mathbb{C}^{n\times n}$ be a bounded set.
                  Denote by $\Sigma^m$ the set of all products of $m$
                  matrices from $\Sigma$ and by $\Sigma'$ -- the semigroup,
                  generated by $\Sigma$. The set $\Sigma$ is said to be
                  asymptotically stable if there is a constant
                  $\alpha\in(0,1)$, such that there are bounded neighborhoods
                  $U,V\subset\mathbb{C}^n$ of the origin for which
                  $AV\subset\alpha^{m}U$ for all $A\in\Sigma^m$,
                  $m=1,2,\ldots$ (i.e. if there exists a norm $|\cdot|$ in
                  $\mathbb{C}^n$ such that $|A|\leq\alpha<1$ for all
                  $A\in\Sigma$). It is shown that the asymptotic stability of
                  $\Sigma$ is equivalent to any of the following three
                  conditions: (i) $\widehat{\rho}(\Sigma)=
                  \limsup_{m\to\infty} (\sup_{A\in\Sigma^m} |A|)^{1/m}<1$,
                  (ii) $\rho(\Sigma)= \limsup_{m\to\infty}
                  (\sup_{A\in\Sigma^m}\rho(A))^{1/m}<1$, (iii) there exists
                  $\alpha>0$, such that $\rho(A)\leq\alpha<1$ for all
                  $A\in\Sigma'$, where $\rho(A)$ is the spectral radius of
                  $A$. The generalized Gelfand spectral radius formula
                  $\rho(\Sigma)= \widehat{\rho}(\Sigma)$ follows immediately
                  from the above results.",
}

@ARTICLE{SWMWK:SIAMREV07,
 author        = "Shorten, Robert and Wirth, Fabian and Mason, Oliver and
                  Wulff, Kai and King, Christopher",
 title         = "Stability criteria for switched and hybrid systems",
 journal       = "SIAM Rev.",
 fjournal      = "SIAM Review",
 year          = "2007",
 volume        = "49",
 number        = "4",
 pages         = "545--592",
 issn          = "0036-1445",
 mrclass       = "93D30 (34A37 34D08 34D20)",
 mrnumber      = "2375524 (2009e:93146)",
 zblnumber     = "1127.93005",
 doi           = "10.1137/05063516X",
 url           = "https://epubs.siam.org/doi/10.1137/05063516X",
 annote        = "The study of the stability properties of switched and hybrid
                  systems gives rise to a number of interesting and
                  challenging mathematical problems. The objective of this
                  paper is to outline some of these problems, to review
                  progress made in solving them in a number of diverse
                  communities, and to review some problems that remain open.
                  An important contribution of our work is to bring together
                  material from several areas of research and to present
                  results in a unified manner. We begin our review by
                  relating the stability problem for switched linear systems
                  and a class of linear differential inclusions. Closely
                  related to the concept of stability are the notions of
                  exponential growth rates and converse Lyapunov theorems,
                  both of which are discussed in detail. In particular,
                  results on common quadratic Lyapunov functions and
                  piecewise linear Lyapunov functions are presented, as they
                  represent constructive methods for proving stability and
                  also represent problems in which significant progress has
                  been made. We also comment on the inherent difficulty in
                  determining stability of switched systems in general, which
                  is exemplified by NP-hardness and undecidability results.
                  We then proceed by considering the stability of switched
                  systems in which there are constraints on the switching
                  rules, through both dwell-time requirements and
                  state-dependent switching laws. Also in this case the
                  theory of Lyapunov functions and the existence of converse
                  theorems are reviewed. We briefly comment on the classical
                  Lur'e problem and on the theory of stability radii, both of
                  which contain many of the features of switched systems and
                  are rich sources of practical results on the topic. Finally
                  we present a list of questions and open problems which
                  provide motivation for continued research in this area.",
}

@MISC{ShulTur:ArXiv12,
 author        = "Shulman, V. S. and Turovskii, {\relax{}Yu}. V.",
 authauthor    = "Shulman, V. and Turovskii, {\relax{}Yu}.",
 title         = "Topological radicals, {III}. {J}oint spectral radius",
 howpublished  = "ArXiv.org e-Print archive",
 year          = "2012",
 month         = aug,
 eprinttype    = "arXiv",
 eprint        = "1208.4592",
 doi           = "10.48550/arXiv.1208.4592",
 note          = "In Russian",
 annote        = "We develop the theory of topological radicals, and in
                  particular the theory of the hypocompact radical. The
                  results are applied for obtaining convenient formulas of
                  calculation of the joint spectral radius of a precompact
                  family of elements in a Banach algebra.",
}

@MISC{ShulTur:ArXiv08,
 author        = "Shulman, Victor S. and Turovskii, {\relax{}Yu}. V.",
 authauthor    = "Shulman, Victor and Turovskii, {\relax{}Yu}.",
 title         = "Application of topological radicals to calculation of joint
                  spectral radii",
 howpublished  = "ArXiv.org e-Print archive",
 year          = "2008",
 month         = may,
 eprinttype    = "arXiv",
 eprint        = "0805.0209",
 doi           = "10.48550/arXiv.0805.0209",
 annote        = "It is shown that the joint spectral radius $\rho(M)$ of a
                  precompact family $M$ of operators on a Banach space $X$ is
                  equal to the maximum of two numbers: the joint spectral
                  radius $\rho_{e}(M)$ of the image of $M$ in the Calkin
                  algebra and the Berger--Wang radius $r(M)$ defined by the
                  formula \[r(M)=\underset{n\rightarrow\infty}{\limsup}\left(
                  \sup\left\{ \rho(a):a\in M^{n}\right\} ^{1/n}\right).\]
                  Some more general Banach-algebraic results of this kind are
                  also proved. The proofs are based on the study of special
                  radicals on the class of Banach algebras.",
}

@ARTICLE{ShulTur:JFA00,
 author        = "Shulman, Victor S. and Turovski{\u\i}, {\relax{}Yu}ri{\u\i}
                  V.",
 authauthor    = "V. Shulman and {\relax{}Yu}. Turovskii",
 title         = "Joint spectral radius, operator semigroups, and a problem of
                  {W}. {W}ojty{\'n}ski",
 journal       = "J. Funct. Anal.",
 fjournal      = "Journal of Functional Analysis",
 year          = "2000",
 volume        = "177",
 number        = "2",
 pages         = "383--441",
 coden         = "JFUAAW",
 issn          = "0022-1236",
 mrclass       = "47L10 (17B65 47A15 47D03 47L70)",
 mrnumber      = "1795957 (2002d:47099)",
 mrreviewer    = "P. Rosenthal",
 zblnumber     = "0980.47008",
 zblreviewer   = "Khristo N.Boyadzhiev (Ada)",
 doi           = "10.1006/jfan.2000.3640",
 url           = "https://www.sciencedirect.com/science/article/pii/S0022123600936401",
 annote        = "Recently, the second author proved a remarkable theorem:
                  \emph{Every semigroup of compact quasinilpotent (i.e.
                  Volterra) Banach space operators has a nontrivial invariant
                  subspace}~\cite{Turovskii:JFA99}. The essence of the proof
                  is that the algebra generated by the semigroup also
                  consists of quasinilpotent operators. Algebras of compact
                  quasinilpotent operators have invariant subspaces,
                  according to V.\,I.~Lomonosov's theory. The main tool used
                  by Turovskii was the joint spectral radius technique
                  developed by him.\par The present paper continues the
                  research on the connection between invariant subspace
                  theory and the joint spectral radius technique.\par
                  W.~Wojty{\'n}ski [Stud. Math. 49, 263--273 (1977)] showed
                  that locally finite Lie algebras of compact operators have
                  invariant subspaces. The question remained open for general
                  Lie algebras of Volterra operators. This question is
                  answered in the affirmative here.\par The authors use
                  semigroups of principle operators (operators whose spectral
                  radius coincides with their essential spectral radius). It
                  is shown that if one such semigroup contains the group
                  $\left\{e^{tT}, t\in\mathbb{R}\right\}$ for some nonzero
                  Volterra operator $T$, then it has a hyperinvariant
                  subspace. This gives the existence of hyperinvariant
                  subspaces for Lie algebras of Volterra operators.\par The
                  paper contains many accompanying results which are of
                  independent interest. In particular, it is proved that the
                  joint spectral radius is a subharmonic function: let $D$ be
                  a domain on the complex plane $\mathbb{C}$ and $\lambda\in
                  D\to M(\lambda)$ be a map into the set of bounded subsets
                  of a Banach algebra $A$. The map $\lambda\to M(\lambda)$ is
                  said \emph{to be analytic} if there exists a family $F$ of
                  analytic function $f: D\to A$ such that
                  $M(\lambda)=\{f(\lambda): f \in F\}$ for all $\lambda\in
                  D$. The family $F$ is called \emph{an analytic family for
                  the map} $\lambda\to M(\lambda)$. We say that $F$ is
                  continuous at a point $\lambda_0 \in D$ if $\sup\{\|f
                  (\lambda)- f(\lambda_0)\| : f \in F\}\to 0$ under
                  $\lambda\to 0$, $\lambda\in D$. Also, $F$ is
                  \emph{continuous} on $D$ if it is continuous at every point
                  of $D$. Sometimes it is written $F(\lambda)$ instead of
                  $M(\lambda)$.\par \textbf{Theorem 3.5.} \emph{Let
                  $\lambda\to M(\lambda)$, $\lambda\in D$, be an analytic map
                  into the set of bounded subsets of $A$ with analytic family
                  $F$. If $F$ is continuous on $D$ then the functions
                  $\lambda\to\log \rho(M(\lambda))$ and $\lambda\to
                  \rho(M(\lambda))$ are subharmonic on $D$.}",
}

@ARTICLE{ShulTur:SM02,
 author        = "Shulman, Victor S. and Turovski{\u\i}, {\relax{}Yu}ri{\u\i}
                  V.",
 authauthor    = "V. Shulman and {\relax{}Yu}. Turovskii",
 title         = "Formulae for joint spectral radii of sets of operators",
 journal       = "Studia Math.",
 fjournal      = "Studia Mathematica",
 year          = "2002",
 volume        = "149",
 number        = "1",
 pages         = "23--37",
 issn          = "0039-3223",
 mrclass       = "47A13 (47A30 47D03)",
 mrnumber      = "1881714 (2002k:47016)",
 mrreviewer    = "Aleksander S. Fain{\v{s}}tein",
 zblnumber     = "1016.47006",
 zblreviewer   = "Pavel Dyshlovenko (Ulyanovsk)",
 doi           = "10.4064/sm149-1-2",
 url           = "https://www.impan.pl/en/publishing-house/journals-and-series/studia-mathematica/all/149/1/90878/formulae-for-joint-spectral-radii-of-sets-of-operators",
 annote        = "The formula $\rho(M)=\max\{\rho_{\chi}(M),r(M)\}$ is proved
                  for precompact sets $M$ of weakly compact operators on a
                  Banach space. Here $\rho(M)$ is the joint spectral radius
                  (the Rota--Strang radius), $\rho_{\chi}(M)$ is the
                  Hausdorff spectral radius (connected with the Hausdorff
                  measure of noncompactness) and $r(M)$ is the Berger--Wang
                  radius. The work is a further development of the ideas of
                  earlier papers, namely \emph{G.-C.~Rota} and
                  \emph{G.~Strang}~\cite{RotaStr:IM60} and
                  \emph{M.\,A.~Berger} and
                  \emph{Y.~Wang}~\cite{BerWang:LAA92}.",
}

@ARTICLE{SU:SIAMJMAA94,
 author        = "Stanford, David P. and Urbano, Jos{\'e} Miguel",
 authauthor    = "Stanford, David and Urbano, Jos{\'e}",
 title         = "Some convergence properties of matrix sets",
 journal       = "SIAM J. Matrix Anal. Appl.",
 fjournal      = "SIAM Journal on Matrix Analysis and Applications",
 year          = "1994",
 volume        = "15",
 number        = "4",
 pages         = "1132--1140",
 issn          = "0895-4798",
 mrclass       = "15A60 (93D15)",
 mrnumber      = "1293908",
 mrreviewer    = "John Maroulas",
 zblnumber     = "0807.15014",
 zblreviewer   = "J. Zem{\'a}nek",
 doi           = "10.1137/S0895479892228213",
 url           = "https://epubs.siam.org/doi/10.1137/S0895479892228213",
 annote        = "A set $\mathcal{A}=\{A_{j}: j\in J\}$ of $n\times{}n$
                  matrices is pointwise convergent provided each $n$-vector
                  $x$ can be steered to zero by iterated multiplication by
                  matrices in $\mathcal{A}$. The convergence is uniform if
                  the sequence of multipliers may be chosen independently of
                  $x$. This paper discusses conditions related to convergence
                  for sets of diagonal, triangular, and general matrices,
                  real and complex. It generalizes known conditions for
                  convergence of a single matrix and characterizes
                  convergence of a set of diagonal matrices in terms of
                  semipositivity of a matrix derived from the set.",
}

@INCOLLECTION{Strang03,
 author        = "Strang, Gilbert",
 editor        = "Dhombres, Jean and Kung, Joseph P. S. and Starr, Norton",
 title         = "The joint spectral radius",
 booktitle     = "Gian-{C}arlo {R}ota on analysis and probability. {S}elected
                  papers and commentaries",
 series        = "Contemporary Mathematicians",
 publisher     = "Birkh{\"a}user Boston Inc.",
 address       = "Boston, MA",
 year          = "2003",
 pages         = "98--102",
 annote        = "The paper ``A note on the joint spectral
                  radius''~\cite{RotaStr:IM60} gives a natural definition of
                  the joint spectral radius of two matrices $A$ and $B$:\[
                  \rho(A,B) = \liminf_{N\to\infty}~ (\text{largest norm of
                  any product with }N\text{ factors})^{1/N}.\] The product
                  can have $A$'s and $B$'s in any order (and the liminf is
                  actually a limit). I was very proud to be a joint author
                  with Gian-Carlo, but I am now a little ashamed that we
                  never thought seriously about how to compute this number
                  $\rho$. For a single matrix it equals the largest magnitude
                  of the eigenvalues $\lambda(A)$. But products of $A$ and
                  $B$ can produce norms and eigenvalues that are very hard to
                  estimate (as $N$ increases) from the two matrices. The
                  Lyapunov exponent is a similar number, using averages over
                  products of length $N$ instead of maxima, and it suffers
                  from the same difficulty (impossibility?) in actual
                  computation. An equivalent definition of $\rho$ is the
                  infimum over all matrix norms of $\max(\|A\|, \|B\|)$. The
                  definitions extend directly to sets of more than two
                  matrices, and an $l_p$ norm joint spectral radius has also
                  proved useful~\cite{Jia:ACM95,LagWang:LAA95}.\par Every few
                  years, Gian-Carlo would ask me whether anyone ever read our
                  paper. After I had tenure, I could tell him the truth:
                  ``not often''. Part of the reason may have been the
                  relatively unfamiliar journal, and the nonexistence of the
                  Internet, and even the atypically obscure language that he
                  had chosen to express our (very simple) idea. In recent
                  years I could change my answer! The joint spectral radius
                  suddenly found application in wavelet theory, especially in
                  the work of \emph{Ingrid Daubechies} and \emph{Jeff
                  Lagarias}~\cite{DaubLag:LAA92}. Of special interest is the
                  question whether $\rho < 1$, so that products with more and
                  more factors of $A$ and $B$ all go to zero.",
}

@INPROCEEDINGS{SBKK:CDC97,
 author        = "Su, Y.-F. and Bhaya, A. and Kaszkurewicz, E. and Kozyakin,
                  V. S.",
 authauthor    = "Su, Yangfeng and Bhaya, A. and Kaszkurewicz, E. and
                  Kozyakin, V.",
 title         = "Further results on stability of asynchronous discrete-time
                  linear systems",
 booktitle     = "Proc. 36th IEEE Conf. Decision and Control",
 year          = "1997",
 volume        = "1",
 pages         = "915--920",
 doi           = "10.1109/CDC.1997.650759",
 url           = "https://ieeexplore.ieee.org/document/650759",
 annote        = "Focuses on the stability problem of discrete-time
                  asynchronous linear systems, which can be viewed as linear
                  systems with time-varying delays. Within this context, a
                  stronger version of the necessity part of the classical
                  Chazan--Miranker theorem~\cite{ChM:LAA69} is proved and new
                  results for special classes of system matrices are also
                  presented.",
}

@ARTICLE{SBKK:LAA98,
 author        = "Su, Yangfeng and Bhaya, Amit and Kaszkurewicz, Eugenius and
                  Kozyakin, Victor S.",
 authauthor    = "Su, Yangfeng and Bhaya, Amit and Kaszkurewicz, Eugenius and
                  Kozyakin, Victor",
 title         = "Further results on convergence of asynchronous linear
                  iterations",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "1998",
 volume        = "281",
 number        = "1-3",
 pages         = "11--24",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "65F30 (65F10)",
 mrnumber      = "1645319 (99h:65087)",
 mrreviewer    = "W. Govaerts",
 zblnumber     = "0935.65021",
 doi           = "10.1016/S0024-3795(98)10030-7",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379598100307",
 annote        = "This paper focuses on the convergence problem of
                  asynchronous linear iterations. A stronger version of the
                  necessity part of the theorem of \emph{D.~Chazan} and
                  \emph{W.~Miranker}~\cite{ChM:LAA69} is proved and new
                  results for special classes of iteration matrices are also
                  presented.",
}

@ARTICLE{Sun:IEEETAC08,
 author        = "Sun, Zhendong",
 title         = "A note on marginal stability of switched systems",
 journal       = "IEEE Trans. Automat. Control",
 fjournal      = "Institute of Electrical and Electronics Engineers.
                  Transactions on Automatic Control",
 year          = "2008",
 volume        = "53",
 number        = "2",
 pages         = "625--631",
 issn          = "0018-9286",
 mrclass       = "93E15",
 mrnumber      = "2394405",
 zblnumber     = "1367.93706",
 doi           = "10.1109/TAC.2008.917644",
 url           = "https://ieeexplore.ieee.org/document/4471856",
 annote        = "In this note, we present criteria for marginal stability and
                  marginal instability of switched systems. For switched
                  nonlinear systems, we prove that uniform stability is
                  equivalent to the existence of a common weak Lyapunov
                  function (CWLF) that is generally not continuous. For
                  switched linear systems, we present a unified treatment for
                  marginal stability and marginal instability for both
                  continuous-time and discrete-time switched systems. In
                  particular, we prove that any marginally stable system
                  admits a norm as a CWLF. By exploiting the largest
                  invariant set contained in a polyhedron, several insightful
                  algebraic characteristics are revealed for marginal
                  stability and marginal instability.",
}

@ARTICLE{Szep:AMH86,
 author        = "Sz{\'e}p, Gabriella",
 title         = "Simultaneous triangularization of projector matrices",
 journal       = "Acta Math. Hungar.",
 fjournal      = "Acta Mathematica Hungarica",
 year          = "1986",
 volume        = "48",
 number        = "3-4",
 pages         = "285--288",
 issn          = "0236-5294",
 mrclass       = "15A21",
 mrnumber      = "861844 (87k:15016)",
 mrreviewer    = "Thomas L. Markham",
 zblnumber     = "0613.15009",
 zblreviewer   = "P. Reichensperger",
 doi           = "10.1007/BF01951353",
 url           = "https://link.springer.com/article/10.1007/BF01951353",
 annote        = "It is known that every matrix can be transformed into upper
                  triangular form with unitary transformation. Now consider
                  the problem of simultaneous triangularization of two
                  matrices. Considering the results mentioned by H.~Shapiro
                  it seems to be useful to find simple necessary and
                  sufficient conditions which can be easily verified for
                  special matrices to be simultaneously triangularisable with
                  unitary transformation.\par \textbf{Theorem.} \emph{Assume
                  that $C$ and $D$ are projector matrices with complex
                  elements. Then they can be simultaneously transformed into
                  upper triangular form with unitary transformation if and
                  only if $CD-DC$ is nilpotent.}",
}

@TECHREPORT{Szyld:TU98,
 author        = "Szyld, Daniel B.",
 authauthor    = "Szyld, Daniel",
 title         = "The Mistery of Asynchronous Iterations Convergence when the
                  Spectral Radius is One",
 institution   = "College of Science and Technology",
 address       = "Temple University",
 year          = "1998",
 number        = "98--102",
 type          = "Report",
 url           = "https://math.temple.edu/~szyld/reports/mystery.pdf",
 annote        = "\emph{D.~Chazan} and \emph{W.~Miranker}~\cite{ChM:LAA69},
                  and other authors since, have shown that a necessary and
                  sufficient condition for asynchronous iterations to
                  converge is that the spectral radius of the absolute value
                  of the iteration matrix is strictly less than one.
                  Nevertheless, several authors, including
                  \emph{B.~Lubachevsky} and
                  \emph{D.~Mitra}~\cite{LubMit:JACM86}, show convergence for
                  asynchronous iterations for singular matrices and matrices
                  representing Markov chains when the spectral radius of the
                  nonnegative iteration matrix is exactly one. In this note,
                  this apparent contradiction is resolved. It is shown that
                  in fact, the spectral radius less than one is a sufficient
                  condition. The necessity is a more subtle issue. Under
                  certain conditions, with spectral radius equal to one,
                  convergence can indeed be achieved.",
}

@ARTICLE{TeiMarg:Autom12,
 author        = "Teichner, Ron and Margaliot, Michael",
 title         = "Explicit construction of a {B}arabanov norm for a class of
                  positive planar discrete-time linear switched systems",
 journal       = "Automatica J. IFAC",
 fjournal      = "Automatica. A Journal of IFAC, the International Federation
                  of Automatic Control",
 year          = "2012",
 volume        = "48",
 number        = "1",
 pages         = "95--101",
 coden         = "ATCAA9",
 issn          = "0005-1098",
 mrclass       = "93D20 (93C55 93D09)",
 mrnumber      = "2879415 (2012m:93354)",
 mrreviewer    = "Ma. Isabel Garc{\'\i}a-Planas",
 zblnumber     = "1244.93092",
 doi           = "10.1016/j.automatica.2011.09.028",
 url           = "https://www.sciencedirect.com/science/article/pii/S0005109811004687",
 annote        = "We consider the stability under arbitrary switching of a
                  discrete-time linear switched system. A powerful approach
                  for addressing this problem is based on studying the ``Most
                  Unstable'' Switching Law (MUSL). If the solution of the
                  switched system corresponding to the MUSL converges to the
                  origin, then the switched system is stable for any
                  switching law. The MUSL can be characterized using optimal
                  control techniques. This variational approach leads to a
                  Hamilton--Jacobi--Bellman equation describing the behavior
                  of the switched system under the MUSL. The solution of this
                  equation is sometimes referred to as a Barabanov norm of
                  the switched system. Although the Barabanov norm was
                  studied extensively, it seems that there are few examples
                  where it was actually computed in closed-form. In this
                  paper, we consider a special class of positive planar
                  discrete-time linear switched systems and provide a
                  closed-form expression for a corresponding Barabanov norm
                  and a MUSL. The unit circle in this norm is a
                  parallelogram.",
}

@PHDTHESIS{Theys:PhD05,
 author        = "Theys, Jacques",
 title         = "Joint Spectral Radius: {T}heory and Approximations",
 school        = "Facult{\'e} des sciences appliqu{\'e}es, D{\'e}partement
                  d'ing{\'e}nierie math{\'e}matique, Center for Systems
                  Engineering and Applied Mechanics",
 address       = "Universit{\'e} Catholique de Louvain",
 year          = "2005",
 pagetotal     = "189",
 month         = may,
 url           = "https://dial.uclouvain.be/pr/boreal/object/boreal:5161",
 annote        = "The spectral radius of a matrix is a widely used concept in
                  linear algebra. It expresses the asymptotic growth rate of
                  successive powers of the matrix. This concept can be
                  extended to sets of matrices, leading to the notion of
                  ``joint spectral radius''. The joint spectral radius of a
                  set of matrices was defined in the 1960's, but has only
                  been used extensively since the 1990's. This concept is
                  useful to study the behavior of multi-agent systems, to
                  determine the continuity of wavelet basis functions or for
                  expressing the capacity of binary codes. Although the joint
                  spectral radius shares some properties with the usual
                  spectral radius, it is much harder to compute, and the
                  problem of approximating it is NP-hard. In this thesis, we
                  first review theoretical results that lead to basic
                  approximations of the joint spectral radius. Then, we list
                  various specific cases where it is effectively computable,
                  before presenting a specific type of sets of matrices, for
                  which we solve the problem of computing it with a
                  polynomial computational cost.",
}

@MISC{Thomas:ArXiv11,
 author        = "Thomas, Alain",
 title         = "Can an infinite left-product of nonnegative matrices be
                  expressed in terms of infinite left-products of stochastic
                  ones?",
 howpublished  = "ArXiv.org e-Print archive",
 year          = "2009",
 month         = aug,
 eprinttype    = "arXiv",
 eprint        = "0908.3538",
 doi           = "10.48550/arXiv.0908.3538",
 annote        = "If a left-product $M_n\cdots M_1$ of square complex matrices
                  converges to a nonnull limit when $n\to\infty$ and if the
                  $M_n$ belong to a finite set, it is clear that there exists
                  an integer $n_0$ such that the $M_n$, $n\ge n_0$, have a
                  common right-eigenvector $V$ for the eigenvalue~$1$. Now
                  suppose that the $M_n$ are nonnegative and that $V$ has
                  positive entries. Denoting by $\Delta$ the diagonal matrix
                  whose diagonal entries are the entries of $V$, the
                  stochastic matrices $S_n=\Delta^{-1}M_n\Delta$ satisfy
                  $M_n\cdots M_{n_0}=\Delta S_n\cdots S_{n_0}\Delta^{-1}$, so
                  the problem of the convergence of $M_n\cdots M_1$ reduces
                  to the one of $S_n\cdots S_{n_0}$. In this paper we still
                  suppose that the $M_n$ are nonnegative but we do not
                  suppose that $V$ has positive entries. The first section
                  details the case of the $2\times2$ matrices, and the last
                  gives a first approach in the case of $d\times d$
                  matrices.",
}

@MISC{Thomas:ArXiv10,
 author        = "Thomas, Alain",
 title         = "Infinite products of nonnegative $2\times 2$ matrices by
                  nonnegative vectors",
 howpublished  = "ArXiv.org e-Print archive",
 year          = "2010",
 month         = jun,
 eprinttype    = "arXiv",
 eprint        = "1006.4050",
 doi           = "10.48550/arXiv.1006.4050",
 annote        = "Given a finite set $\{M_0,\dots,M_{d-1}\}$ of nonnegative
                  $2\times 2$ matrices and a nonnegative column-vector $V$,
                  we associate to each $(\omega_n)\in\{0,\dots,d-1\}^\mathbb
                  N$ the sequence of the column-vectors
                  $\frac{M_{\omega_1}\dots M_{\omega_n}V}{\Vert
                  M_{\omega_1}\dots M_{\omega_n}V\Vert}$. We give the
                  necessary and sufficient condition on the matrices $M_k$
                  and the vector $V$ for this sequence to converge for all
                  $(\omega_n)\in\{0,\dots,d-1\}^\mathbb N$ such that $\forall
                  n,\ M_{\omega_1}\dots
                  M_{\omega_n}V\ne\left(\begin{smallmatrix}0\\
                  0\end{smallmatrix}\right)$.",
}

@MISC{Thomas:ArXiv13,
 author        = "Thomas, Alain",
 title         = "Almost sure convergence of products of $2\times 2$
                  nonnegative matrices",
 howpublished  = "ArXiv.org e-Print archive",
 year          = "2013",
 month         = feb,
 eprinttype    = "arXiv",
 eprint        = "1302.4715",
 doi           = "10.48550/arXiv.1302.4715",
 annote        = "We study the almost sure convergence of the normalized
                  columns in an infinite product of nonnegative matrices, and
                  the almost sure rank one property of its limit points.
                  Given a probability on the set of $2\times2$ nonnegative
                  matrices, with finite support $\mathcal
                  A=\{A(0),\dots,A(s-1)\}$, and assuming that at least one of
                  the $A(k)$ is not diagonal, the normalized columns of the
                  product matrix $P_n=A(\omega_1)\dots A(\omega_n)$ converge
                  almost surely (for the product probability) with an
                  exponential rate of convergence if and only if the Lyapunov
                  exponents are almost surely distinct. If this condition is
                  satisfied, given a nonnegative column vector $V$ the column
                  vector $\frac{P_nV}{\Vert P_nV\Vert}$ also converges almost
                  surely with an exponential rate of convergence. On the
                  other hand if we assume only that at least one of the
                  $A(k)$ do not have the form
                  $\begin{pmatrix}a&0\\0&d\end{pmatrix}$, $ad\ne0$, nor the
                  form $\begin{pmatrix}0&b\\d&0\end{pmatrix}$, $bc\ne0$, the
                  limit-points of the normalized product matrix
                  $\frac{P_n}{\Vert P_n\Vert}$ have almost surely rank~$1$
                  --~although the limits of the normalized columns can be
                  distinct~-- and $\frac{P_nV}{\Vert P_nV\Vert}$ converges
                  almost surely with a rate of convergence that can be
                  exponential or not exponential.",
}

@MISC{Thomas:Arxiv18,
 author        = "Thomas, Alain",
 title         = "Normalized image of a vector by an infinite product of
                  nonnegative matrices",
 howpublished  = "ArXiv.org e-Print archive",
 year          = "2018",
 month         = aug,
 eprinttype    = "arXiv",
 eprint        = "1808.09803",
 hal_id        = "hal-01857355",
 hal_version   = "v2",
 doi           = "10.48550/arXiv.1808.09803",
 annote        = "Let $P_n=A_1\cdots A_n$, where $\mathcal
                  A=(A_n)_{n\in\mathbb N}$ is a sequence of $d\times d$
                  matrices, and let $V$ be a $d$-dimensional column-vector.
                  We call ``normalized image of~$V$ by the infinite product
                  of the matrices $A_n$'', the vector
                  $\lim_{n\to\infty}P_nV/{\Vert P_nV\Vert}$ if exists. In
                  general the sequence of matrices $n\mapsto P_n/{\Vert
                  P_n\Vert}$ does not converge but, under some sufficient
                  conditions specified in Theorem~A the sequence of vectors
                  $n\mapsto P_nV/{\Vert P_nV\Vert}$ converges. We use
                  Theorem~A to prove that certain sofic (i.e. linearly
                  representable) measures satisfy the multifractal
                  formalism.",
}

@MISC{Thomas:ArXiv22,
 author        = "Thomas, Alain",
 title         = "Normalized image of a vector by an infinite product of
                  nonnegative matrices",
 howpublished  = "ArXiv.org e-Print archive",
 year          = "2022",
 month         = may,
 eprinttype    = "arXiv",
 eprint        = "2205.09044",
 doi           = "10.48550/arXiv.2205.09044",
 annote        = "Let $\nu$ be a measure on the set $\{0,1,\dots,b-1\}^\mathbb
                  N$, linearly representable by means of a finite set
                  $\mathcal M$ of square nonnegative matrices. To prove that
                  $\nu$ has the weak Gibbs property, one generally use the
                  uniform convergence on $\mathcal M^\mathbb N$ of the
                  sequence of vectors $n\mapsto\frac{A_1\cdots A_nc}{\Vert
                  A_1\cdots A_nc\Vert}$, where $c$ is a positive
                  column-vector. We give a sufficient condition for this
                  sequence to converge, that we apply to some measures
                  defined by Bernoulli convolution. Secondarily we prove that
                  the sequence of matrices $n\mapsto\frac{A_1\cdots
                  A_n}{\Vert A_1\cdots A_n\Vert}$ in general diverges.",
}

@ARTICLE{Toker:AJIFAC97,
 author        = "Toker, Onur",
 title         = "On the complexity of the robust stability problem for linear
                  parameter varying systems",
 journal       = "Automatica J. IFAC",
 fjournal      = "Automatica. A Journal of IFAC, the International Federation
                  of Automatic Control",
 year          = "1997",
 volume        = "33",
 number        = "11",
 pages         = "2015--2017",
 coden         = "ATCAA9",
 issn          = "0005-1098",
 mrclass       = "93D09 (65Y20)",
 mrnumber      = "1486901",
 zblnumber     = "0914.93046",
 zblreviewer   = "A. Bacciotti (Torino)",
 doi           = "10.1016/S0005-1098(97)00129-5",
 url           = "https://www.sciencedirect.com/science/article/pii/S0005109897001295",
 annote        = "In this paper, it is shown that the problem of checking
                  robust stability of linear parameter varying (LPV) systems
                  is NP-hard, and therefore, it is rather unlikely to find
                  polynomial-time solution procedures for this problem. In
                  the frequency-domain structured uncertainty case, it is
                  known that the robust stability problem is NP-hard (Toker
                  and {\"O}zbay, 1995; Token 1995; Poljak and Rohn, 1993;
                  Nemirovski, 1993; Braatz et al., 1994), but allowing the
                  uncertain blocks to be time varying gives a computationally
                  tractable problem (Shamma, 1994; Poolla and Tikku, 1995),
                  which can be solved by convex optimization techniques. In
                  the parametric uncertainty case, NP-hardness of the robust
                  stability problem has been shown in (Poljak and Rohn, 1993;
                  Nemirovski, 1993). The results of this paper show that,
                  allowing the uncertain parameters to be time varying, does
                  not give a computationally simpler problem, i.e. it remains
                  NP-hard, and hence it is rather unlikely to find
                  computationally tractable solution procedures for this
                  problem. On the other hand, as far as the existence of an
                  algorithm is considered, there is still no known
                  (non-polynomial time) algorithm for the robust stability
                  problem of linear parameter varying systems (Lagarias and
                  Wang, 1995), and the well-known Tarski's theorem (Tarski,
                  1951) does not provide a solution procedure (Kozyakin,
                  1990). Recently, there has been some developments in the
                  direction of constructing non-polynomialtime algorithms for
                  a related problem, called the joint spectral radius (JSR)
                  computation problem (Lagarias and Wang, 1995). We also
                  comment on the use of these results for developing a
                  non-polynomial-time algorithm for testing robust stability
                  of linear parameter varying systems.\par The author
                  considers linear parameter varying systems
                  \[x(k+1)=\left(A_0 + \sum_{i=1}^{n} r_{i}(k) A_{i}\right)
                  x(k),\] where $\{r_{i}(k)\}$ is a sequence such that
                  $\|r_{i}(k)\|_{\infty}\le1$ for each $i=1,\ldots,n$. The
                  system is said to be stable if for each choice of the
                  sequences $\{r_{i}(k)\}$ and each initial condition the
                  solution is bounded. The main result is that the problem of
                  checking stability for such a system is NP-hard.",
}

@INPROCEEDINGS{TW:CDC12,
 author        = "Trenn, Stephan and Wirth, Fabian",
 title         = "Linear switched {DAE}s: {L}yapunov exponents, a converse
                  {L}yapunov theorem, and {B}arabanov norms",
 booktitle     = "51st IEEE Conference on Decision and Control (CDC), 10-13
                  Dec., Maui, HI, USA",
 year          = "2012",
 pages         = "2666--2671",
 doi           = "10.1109/CDC.2012.6426245",
 url           = "https://ieeexplore.ieee.org/document/6426245",
 annote        = "For linear switched differential algebraic equations (DAEs)
                  we consider the problem of characterizing the maximal
                  exponential growth rate of solutions. It is shown that a
                  finite exponential growth rate exists if and only if the
                  set of consistency projectors associated to the family of
                  DAEs is product bounded. This result may be used to derive
                  a converse Lyapunov theorem for switched DAEs. Under the
                  assumption of irreducibility we show that a construction
                  reminiscent of the construction of Barabanov norms is
                  feasible as well.",
}

@ARTICLE{TB:MCSS97-4,
 author        = "Tsitsiklis, John N. and Blondel, Vincent D.",
 authauthor    = "Tsitsiklis, John and Blondel, Vincent",
 title         = "Lyapunov exponents of pairs of matrices. {A} correction:
                  ``{T}he {L}yapunov exponent and joint spectral radius of
                  pairs of matrices are hard --- when not impossible --- to
                  compute and to approximate''",
 journal       = "Math. Control Signals Systems",
 fjournal      = "Mathematics of Control, Signals, and Systems",
 year          = "1997",
 volume        = "10",
 number        = "4",
 pages         = "381",
 coden         = "MCSYE8",
 issn          = "0932-4194",
 mrclass       = "65Y20 (15A42 34D08 68Q25)",
 mrnumber      = "1486730 (99h:65238b)",
 doi           = "10.1007/BF01211553",
 url           = "https://link.springer.com/article/10.1007/BF01211553",
}

@ARTICLE{TB:MCSS97-1,
 author        = "Tsitsiklis, John N. and Blondel, Vincent D.",
 authauthor    = "Tsitsiklis, John and Blondel, Vincent",
 title         = "The {L}yapunov exponent and joint spectral radius of pairs
                  of matrices are hard --- when not impossible --- to compute
                  and to approximate",
 journal       = "Math. Control Signals Systems",
 fjournal      = "Mathematics of Control, Signals, and Systems",
 year          = "1997",
 volume        = "10",
 number        = "1",
 pages         = "31--40",
 coden         = "MCSYE8",
 issn          = "0932-4194",
 mrclass       = "65Y20 (15A42 34D08 68Q25)",
 mrnumber      = "1462278 (99h:65238a)",
 zblnumber     = "0888.65044",
 zblreviewer   = "A. L. Andrew (Bundoora)",
 doi           = "10.1007/BF01219774",
 url           = "https://link.springer.com/article/10.1007/BF01219774",
 annote        = "The authors show that the joint spectral radius and the
                  generalized spectral radius of two integer matrices are not
                  approximable in polynomial time, and that two related
                  quantities -- the lower spectral radius and the largest
                  Lyapunov exponent -- are not algorithmically approximable.
                  Some applications of these results are discussed.",
}

@ARTICLE{Tuna:AJIFAC08,
 author        = "Tuna, S. Emre",
 authauthor    = "Tuna, Emre",
 title         = "Growth rate of switched homogeneous systems",
 journal       = "Automatica J. IFAC",
 fjournal      = "Automatica. A Journal of IFAC, the International Federation
                  of Automatic Control",
 year          = "2008",
 volume        = "44",
 number        = "11",
 pages         = "2857--2862",
 coden         = "ATCAA9",
 issn          = "0005-1098",
 mrclass       = "93D05 (93C55)",
 mrnumber      = "2527207",
 zblnumber     = "1152.93492",
 doi           = "10.1016/j.automatica.2008.03.017",
 url           = "https://www.sciencedirect.com/science/article/pii/S0005109808002501",
 annote        = "We consider discrete-time homogeneous systems under
                  arbitrary switching and study their growth rate, the
                  analogue of joint spectral radius for switched linear
                  systems. We show that a system is asymptotically stable if
                  and only if its growth rate is less than unity. We also
                  provide an approximation algorithm to compute growth rate
                  with arbitrary accuracy.",
}

@ARTICLE{Turovskii:JFA99,
 author        = "Turovskii, {\relax{}Yu}. V.",
 authauthor    = "Turovskii, {\relax{}Yu}.",
 title         = "Volterra semigroups have invariant subspaces",
 journal       = "J. Funct. Anal.",
 fjournal      = "Journal of Functional Analysis",
 year          = "1999",
 volume        = "162",
 number        = "2",
 pages         = "313--322",
 coden         = "JFUAAW",
 issn          = "0022-1236",
 mrclass       = "47A15 (47D03)",
 mrnumber      = "1682061 (2000d:47017)",
 mrreviewer    = "P. Rosenthal",
 zblnumber     = "0921.47008",
 zblreviewer   = "P. Dyshlovenko (Ul'yanovsk)",
 doi           = "10.1006/jfan.1998.3368",
 url           = "https://www.sciencedirect.com/science/article/pii/S0022123698933687",
 annote        = "The article is devoted to the solution of the Volterra
                  semigroup problem. This problem was raised by E. Nordgren,
                  H. Radjavi and P. Rosenthal [Indiana Univ. Math. J. 33,
                  271--275 (1984)] and has the following form: does every
                  (multiplicative) semigroup of compact quasinilpotent
                  operators on an infinite dimensional Banach space have a
                  nontrivial invariant subspace?\par The problem is solved
                  with no additional conditions related to nuclearity,
                  positivity, special spectral behavior and so on. Some
                  consequences are also considered.",
}

@ARTICLE{TurShul:FAP00,
 author        = "Turovskii, {\relax{}Yu}. V. and Shul'man, V. S.",
 authauthor    = "Turovskii, {\relax{}Yu}. and Shulman, V.",
 title         = "Joint spectral radius and invariant subspaces",
 journal       = "Funct. Anal. Appl.",
 fjournal      = "Functional Analysis and its Applications",
 year          = "2000",
 volume        = "34",
 number        = "2",
 pages         = "156--158",
 issn          = "0016-2663",
 mrclass       = "47A13 (47A15 47D03)",
 mrnumber      = "1773855 (2002b:47014)",
 mrreviewer    = "Hari Bercovici",
 zblnumber     = "0978.47004",
 zblreviewer   = "Michal Zajac (Bratislava)",
 doi           = "10.1007/BF02482436",
 url           = "https://link.springer.com/article/10.1007/BF02482436",
 annote        = "The paper is a review of some recent results of the authors
                  (no proofs are given) on existence of nontrivial invariant
                  and hyperinvariant subspaces for some families of Banach
                  space operators. For a bounded set $M$ of elements of a
                  normed algebra $A$ put $\|M\|=\{\|a\|:a\in M\}$. Let $M^n$
                  be the set of all products of n elements of $M$.
                  $\rho(M)=\inf\|M^n\|^{1/n}$ is called the spectral radius
                  of $M$. The essential spectral radius of $M$ is the
                  spectral radius of the image of $M$ in the Calkin algebra.
                  Most of the results of the paper contain conditions on
                  $\rho(M)$ and/or $\rho_{e}(M)$. The authors gave answers to
                  questions 4 and 5 posed by \emph{W.~Wojtynski} [Stud. Math.
                  59, 263--273 (1977)].",
}

@ARTICLE{TurShul:FAP12,
 author        = "Turovskii, {\relax{}Yu}. V. and Shulman, V. S.",
 authauthor    = "{\relax{}Yu}. Turovskii and V. Shulman",
 title         = "Topological radicals and joint spectral radius",
 journal       = "Funct. Anal. Appl.",
 fjournal      = "Functional Analysis and its Applications",
 year          = "2012",
 volume        = "46",
 number        = "4",
 pages         = "287--304",
 eprinttype    = "arXiv",
 eprint        = "0805.0209",
 issn          = "0016-2663; 1573-8485/e",
 mrclass       = "47A10",
 mrnumber      = "3075096",
 zblnumber     = "1319.46038",
 doi           = "10.1007/s10688-012-0036-y",
 url           = "https://link.springer.com/article/10.1007/s10688-012-0036-y",
 annote        = "It is shown that the joint spectral radius $\rho(M)$ of a
                  precompact family $M$ of operators on a Banach space $X$ is
                  equal to the maximum of two numbers: the joint spectral
                  radius $\rho_{e}(M)$ of the image of $M$ in the Calkin
                  algebra and the Berger--Wang radius $r(M)$ defined by the
                  formula \[r(M)=\underset{n\rightarrow\infty}{\limsup}\left(
                  \sup\left\{ \rho(a):a\in M^{n}\right\} ^{1/n}\right).\]
                  Some more general Banach-algebraic results of this kind are
                  also proved. The proofs are based on the study of special
                  radicals on the class of Banach algebras.",
}

@PHDTHESIS{Vagnoni08,
 author        = "Vagnoni, Cristina",
 title         = "Algorithms for the computation of the joint spectral
                  radius",
 school        = "Dipartimento di Matematica Pura ed Applicata, Scuola di
                  Dottorato di Ricerca in Scienze Matematiche Indirizzo
                  Matematica Computazionale",
 address       = "Universit{\`a} degli Studi di Padova",
 year          = "2008",
 url           = "http://paduaresearch.cab.unipd.it/408/",
 annote        = "The asymptotic behaviour of the solutions of a discrete
                  linear dynamical system is related to the spectral radius
                  $R$ of its associated family $\mathcal{F}$. In particular,
                  a system is stable if $R\le 1$ and there exists an extremal
                  norm for the family $\mathcal{F}$. This kind of systems is
                  important for a large number of applications. In
                  particular, we mention the stability analysis of numerical
                  methods for ordinary differential equations.\par In the
                  last decades some algorithms have been proposed in order to
                  find real extremal norms of polytope type in the case of
                  finite families. However, recently it has been observed
                  that it is more useful to consider complex polytope norms,
                  which are norms whose unit ball is a balanced complex
                  polytope.\par In this work, using the theory developed by
                  \emph{N.~Guglielmi} and \emph{M.~Zennaro}
                  in~\cite{GugZen:JCA07}, we extend the algorithm for the
                  construction (i.e. for the geometric representation) of the
                  unit ball of real polytope norms to the complex space. In
                  order to succeed in our purpose, we first needed to get a
                  deeper theoretical knowledge of the balanced complex
                  polytopes. However, due to the extreme increase in
                  complexity of the geometry of such objects with the
                  dimension n of the space, we have confined ourselves to
                  face the two-dimensional case.\par In particular, we have
                  given original theoretical results on the geometry of
                  two-dimensional balanced complex polytopes in order to
                  present the first two efficient algorithms, one for the
                  construction of a balanced complex polytope in the
                  two-dimensional space and one for the computation of the
                  complex polytope norm of a two-dimensional vector starting
                  from the knowledge of the boundary of the corresponding
                  unit ball.",
}

@MISC{JSRToolbox,
 author        = "Vankeerberghen, G. and Hendrickx, J. and Jungers, R. and
                  Chang, C. T. and Blondel, V.",
 authauthor    = "Vankeerberghen, G. and Hendrickx, J. and Jungers, R. and
                  Chang, Chia-Tche and Blondel, V.",
 title         = "The {JSR} {T}oolbox",
 howpublished  = "{MATLAB}{$^\circledR$} Central",
 year          = "2011",
 url           = "https://www.mathworks.com/matlabcentral/fileexchange/33202-the-jsr-toolbox",
 annote        = "The Joint Spectral Radius of a set of matrices characterizes
                  the maximal asymptotic rate of growth of a product of
                  matrices taken in this set, when the length of the product
                  increases. It is known to be very hard to compute. In
                  recent years, many different methods have been proposed to
                  approximate it. These methods have different advantages,
                  depending on the application considered, the type of
                  matrices considered, the desired accuracy or running time,
                  etc. The goal of this toolbox is to provide the practioner
                  with the best available methods, and propose an easy tool
                  for the researcher to compare the different methods.\par
                  The JSR Toolbox is a MATLAB toolbox that provides a large
                  set of methods for the approximation of the joint spectral
                  radius, but also includes several helper functions, for
                  example for comparison or analysis purposes, and several
                  demonstration functions. One important feature is that the
                  toolbox offers a ``default'' algorithm that computes bounds
                  on the joint spectral radius by combining several
                  approaches presented in this article. This may be useful if
                  one does not know what algorithm is more suitable. The
                  behavior of this algorithm may also be parameterized as
                  needed, for instance by setting a maximal computation time
                  or by fine-tuning a particular step in an algorithm. The
                  main steps of the default algorithm are the following:
                  \begin{itemize} \item Try to transform the problem into a
                  set of smaller independent problems. This is possible when
                  the matrices in the set $\Sigma$ are simultaneously
                  blocktriangularizable. \item If the matrices are
                  nonnegative, start with the pruning algorithm in order to
                  get some bounds $[\beta^-, \beta^+]$ on the joint spectral
                  radius, then compute the joint conic radius, using the
                  positive orthant as cone, and the ellipsoidal norm
                  approximation using $[\beta^-, \beta^+]$ as initial bounds.
                  \item If some matrices have negative entries, start with a
                  variant of Gripenberg's algorithm in order to get some
                  initial bounds and a candidate product. This variant may
                  rescale the matrices during the computation in order to
                  avoid overflows. After this first step, compute the
                  ellipsoidal norm approximation and, if needed, try to
                  certify optimality or to find a better candidate product
                  using a balanced complex polytope method or a conitope
                  method, which is a lifted polytope method. \end{itemize}",
}

@INPROCEEDINGS{VHJ:ACM14,
 author        = "Vankeerberghen, Guillaume and Hendrickx, Julien and Jungers,
                  Rapha{\"e}l M.",
 authauthor    = "Vankeerberghen, Guillaume and Hendrickx, Julien and Jungers,
                  Rapha{\"e}l",
 title         = "{JSR}: {A} Toolbox to Compute the Joint Spectral Radius",
 booktitle     = "Proceedings of the 17th International Conference on Hybrid
                  Systems: Computation and Control",
 series        = "HSCC'14",
 publisher     = "ACM",
 address       = "New York, NY, USA",
 location      = "Berlin, Germany",
 year          = "2014",
 pages         = "151--156",
 pagetotal     = "6",
 isbn          = "978-1-4503-2732-9",
 zblnumber     = "1364.65099",
 acmid         = "2562124",
 doi           = "10.1145/2562059.2562124",
 url           = "https://dl.acm.org/doi/10.1145/2562059.25621244",
 annote        = "We present a toolbox for computing the Joint Spectral Radius
                  of a set of matrices, i.e., the maximal asymptotic growth
                  rate of products of matrices taken in that set. The Joint
                  Spectral Radius has a wide range of applications, including
                  switched and hybrid systems, combinatorial words theory, or
                  the study of wavelets. However, it is notoriously difficult
                  to compute or approximate; it is actually uncomputable, and
                  its approximation is NP-hard. The toolbox compiles several
                  recent computation and approximation methods, and also
                  contains an automatic blackbox method for inexperienced
                  users, selecting the most appropriate methods based on an
                  automatic study of the matrix set provided. The tool is
                  implemented in Matlab and is freely downloadable (with
                  documentation and demos) from Matlab
                  Central~\cite{JSRToolbox}.",
}

@MISC{VarMor:ArXiv22,
 author        = "Varney, Jonah and Morris, Ian D.",
 authauthor    = "Varney, Jonah and Morris, Ian",
 title         = "On marginal growth rates of matrix products",
 howpublished  = "ArXiv.org e-Print archive",
 year          = "2022",
 month         = sep,
 eprinttype    = "arXiv",
 eprint        = "2209.00449",
 doi           = "10.48550/arXiv.2209.00449",
 annote        = "In this article we consider the maximum possible growth rate
                  of sequences of long products of $d \times d$ matrices all
                  of which are drawn from some specified compact set which
                  has been normalised so as to have joint spectral radius
                  equal to $1$. We define the \emph{marginal instability rate
                  sequence} associated to such a set to be the sequence of
                  real numbers whose $n^{th}$ entry is the norm of the
                  largest product of length $n$, and study the general
                  properties of sequences of this form. We describe how new
                  marginal instability rate sequences can be constructed from
                  old ones, extend an earlier example of Protasov and Jungers
                  to obtain marginal instability rate sequences whose limit
                  superior rate of growth matches various non-integer powers
                  of $n$, and investigate the relationship between marginal
                  instability rate sequences arising from finite sets of
                  matrices and those arising from sets of matrices with
                  cardinality $2$. We also give the first example of a finite
                  set whose marginal instability rate sequence is
                  asymptotically similar to a polynomial with non-integer
                  exponent. Previous examples had this property only along a
                  subsequence. ",
}

@PHDTHESIS{Varney21,
 author        = "Varney, Jonah Patric",
 authauthor    = "Varney, Jonah",
 title         = "Marginal instability of matrix systems",
 school        = "Dept. of Mathematics",
 address       = "University of Surrey, Guildford GU2 7XH, United Kingdom",
 year          = "2021",
 doi           = "10.15126/thesis.900392",
 url           = "https://dspace.mit.edu/handle/1721.1/44206",
 annote        = "We study the size of products of matrices taken from a
                  finite set of matrices. In particular we describe the
                  maximal growth rate of the norms of products of matrices
                  from our set. For a set of matrices we define an associated
                  ``FMIRF'' to be the function mapping a natural number n to
                  the maximum of the norm of all possible n length products
                  from our set. Switching systems are models defined by a
                  finite set of matrices, by understanding the associated
                  FMIRF, we gain insight into the stability of the switching
                  system. Switching systems have attracted a lot of interest
                  from control theorists, as they pose a complex challenge
                  and as they have a wide range of applications. Switching
                  systems are needed to describe systems in nature and
                  engineering, because such systems do not fit a single model
                  and exhibit switching between different models depending on
                  various environments. Zhendong and Shuzi have uncovered
                  many tools for analysing nonlinear behaviour in switching
                  systems. We focus on switching systems with FMIRFs that
                  grow slower than exponentially, such FMIRFs are called
                  marginally unstable. We are interested in the different
                  possible rates of growth a marginally unstable system can
                  achieve, significant research by Jungers, Protasov and
                  Blondel in~\cite{JPB:LAA08} showed that in particular cases
                  marginally unstable systems grow like a polynomial. This is
                  not the case though for all systems and in~\cite{PJ:NAHS15}
                  Jungers and Protasov released an example of a FMIRF with
                  sublinear growth. We prove that their example is bounded
                  almost everywhere and adapt it to achieve other sublinear
                  grow rates. We show that if we have a set of two
                  dimensional matrices then the corresponding FMIRF has
                  bounded growth or linear growth. We prove that if we have a
                  set containing a single matrix the the corresponding FMIRF
                  grows like a polynomial. This begs the question, does the
                  cardinality of our set always affect the possible growth of
                  the corresponding FMIRF? We prove that, given a FMIRF
                  associated to finite sets of matrices it is possible to
                  construct a set of two matrices with a FMIRF that has the
                  same growth. We also explore the effect of taking direct
                  and Kronecker products on sets of matrices has on their
                  associated FMIRFs. Finally instead of starting with a set
                  of matrices we start with a function that maps from a
                  dynamical system onto a matrix of $\theta$-H{\"{o}}lder
                  continuous functions. In this more general setting we show
                  how the generalised FMIRF's growth depends on the
                  $\theta$-H{\"{o}}lder continuous functions. An example of a
                  generalised FMIRF with unbounded but sublinear growth is
                  given.",
}

@ARTICLE{VladRub:JCA01,
 author        = "Vladimirov, A. and Rubinov, A.",
 title         = "Dynamics of positive multiconvex relations",
 journal       = "J. Convex Anal.",
 fjournal      = "Journal of Convex Analysis",
 year          = "2001",
 volume        = "8",
 number        = "2",
 pages         = "387--399",
 issn          = "0944-6532",
 mrclass       = "49J53 (47N10 52A41 91B62)",
 mrnumber      = "1915948",
 mrreviewer    = "Diethard Pallaschke",
 zblnumber     = "1042.49027",
 url           = "https://www.heldermann-verlag.de/jca/jca08/jca0207.pdf",
 annote        = "A notion of multiconvex relation as a union of a finite
                  number of convex relations is introduced. For a particular
                  case of multiconvex process, that is, a union of a finite
                  set of convex processes, we define the notions of the joint
                  and the generalized spectral radius in the same manner as
                  for matrices. We prove the equivalence of these two values
                  if all component processes are positive, bounded, and
                  closed.",
}

@MISC{Vlad:ArXiv16,
 author        = "Vladimirov, Alexander",
 title         = "Continuous products of matrices",
 howpublished  = "ArXiv.org e-Print archive",
 year          = "2016",
 month         = mar,
 eprinttype    = "arXiv",
 eprint        = "1603.00854",
 doi           = "10.48550/arXiv.1603.00854",
 annote        = "We answer the question if the continuous product of square
                  matrices $M(t)$ over $t\in [0,1]$ can be correctly defined.
                  The case where all $M(t)$ are taken from a finite set
                  $\Sigma$ is studied. We find necessary and sufficient
                  conditions on $\Sigma$ that ensure the convergence of
                  products $M(t_{0}=0)M(t_{1})\cdots M(t_{N}=1)$ as the
                  partition $0<t_{1}<\dots<1$ refines. These conditions are
                  properties LCP (left convergent product) and RCP (right
                  convergent product) of the set $\Sigma$. That is, it
                  suffices to require the convergence of all finite products
                  $M_{1}M_{2}\cdots M_{K}$ and $M_{K}\cdots M_{2}M_{1}$ as
                  $K\to\infty$, where $M_{i}\in\Sigma$. The theory of joint
                  spectral radius is heavily used.",
}

@ARTICLE{VEB:LAA00,
 author        = "Vladimirov, Alexander and Elsner, Ludwig and Beyn,
                  Wolf-J{\"u}rgen",
 title         = "Stability and paracontractivity of discrete linear
                  inclusions",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2000",
 volume        = "312",
 number        = "1-3",
 pages         = "125--134",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "39A11 (15A60 39A12 93C55 93D05)",
 mrnumber      = "1759327 (2001g:39040)",
 mrreviewer    = "Juan-Miguel Gracia",
 zblnumber     = "0964.15032",
 zblreviewer   = "Michal Zajac (Bratislava)",
 doi           = "10.1016/S0024-3795(00)00094-X",
 url           = "https://www.sciencedirect.com/science/article/pii/S002437950000094X",
 annote        = "A square matrix $A$ is called paracontracting (PC) with
                  respect to a given norm if $Ax\neq x
                  \Rightarrow\|Ax\|<\|x\|$. A set $\Sigma$ of matrices is PC
                  if all elements of $\Sigma$ are PC with respect to the same
                  norm. The main result of this paper is the proof of a
                  conjecture (cf. \emph{W.-J.~Beyn} and
                  \emph{L.~Elsner}~\cite{BE:EJLA97}) that a finite set
                  $\Sigma$ of $n\times n$ matrices is PC iff each infinite
                  trajectory $x_{j+1} =A_j x_j$ ($A_j \in\Sigma$) converges
                  to a point $x^*\in\mathbb{R}^n$, i.e. all infinite products
                  of elements of $\Sigma$ are convergent. Moreover, it is
                  shown that this is equivalent to $\lim_{j\to\infty}
                  \|x_{j+1}-x_{j}\|=0$ and also to the fact that $\Sigma$ is
                  product bounded, i.e. $\exists C>0$ such that $\|A_1\cdots
                  A_m\|<C$ for all finite products of (not necessarily
                  different) elements of $\Sigma$.",
}

@ARTICLE{VlasJung:SCL14,
 author        = "Vlassis, Nikos and Jungers, Rapha{\"e}l",
 title         = "Polytopic uncertainty for linear systems: new and old
                  complexity results",
 journal       = "Systems Control Lett.",
 fjournal      = "Systems {\&} Control Letters",
 year          = "2014",
 volume        = "67",
 pages         = "9--13",
 eprinttype    = "arXiv",
 eprint        = "1310.1930",
 issn          = "0167-6911",
 mrclass       = "93D09 (68Q17 93C41 93D21)",
 mrnumber      = "3183374",
 mrreviewer    = "{\'E}va Gyurkovics",
 zblnumber     = "1288.93056",
 doi           = "10.1016/j.sysconle.2014.02.001",
 url           = "https://www.sciencedirect.com/science/article/pii/S0167691114000292",
 annote        = "We survey the problem of deciding the stability or
                  stabilizability of uncertain linear systems whose region of
                  uncertainty is a polytope. This natural setting has
                  applications in many fields of applied science, from
                  Control Theory to Systems Engineering to Biology.\par We
                  focus on the algorithmic decidability of this property when
                  one is given a particular polytope. This setting gives rise
                  to several different algorithmic questions, depending on
                  the nature of time (discrete/continuous), the property
                  asked (stability/stabilizability), or the type of
                  uncertainty (fixed/switching). Several of these questions
                  have been answered in the literature in the last thirty
                  years. We point out the ones that have remained open, and
                  we answer all of them, except one which we raise as an open
                  question. In all the cases, the results are negative in the
                  sense that the questions are NP-hard.\par As a byproduct,
                  we obtain complexity results for several other matrix
                  problems in Systems and Control.",
}

@INPROCEEDINGS{WangMM:ACC21,
 author        = "Wang, Jie and Maggio, Martina and Magron, Victor",
 title         = "Sparse{JSR}: {A} fast algorithm to compute joint spectral
                  radius via sparse {SOS} decompositions",
 booktitle     = "2021 American Control Conference (ACC)",
 year          = "2021",
 pages         = "2254--2259",
 eprinttype    = "arXiv",
 eprint        = "2008.11441",
 doi           = "10.23919/ACC50511.2021.9483347",
 url           = "https://ieeexplore.ieee.org/document/9483347",
 annote        = "This paper focuses on the computation of joint spectral
                  radii (JSR), when the involved matrices are sparse. We
                  provide a sparse variant of the procedure proposed by
                  Parrilo and Jadbabaie, to compute upper bounds of the JSR
                  by means of sum-of-squares (SOS) relaxations. Our resulting
                  iterative algorithm, called \texttt{SparseJSR}, is based on
                  the \emph{term sparsity} SOS (TSSOS) framework, developed
                  by Wang, Magron and Lasserre, yielding SOS decompositions
                  of polynomials with arbitrary sparse support.
                  \texttt{SparseJSR} exploits the sparsity of the input
                  matrices to significantly reduce the computational burden
                  associated with the JSR computation. Our algorithmic
                  framework is then successfully applied to compute upper
                  bounds for JSR, on randomly generated benchmarks as well as
                  on problems arising from stability proofs of controllers,
                  in relation with possible hardware and software faults.",
}

@INPROCEEDINGS{WangWen:CIS13,
 author        = "Wang, Shuoting and Wen, Jiechang",
 title         = "The Finiteness Conjecture for the Joint Spectral Radius of a
                  Pair of Matrices",
 booktitle     = "Proceedings of the 9th International Conference on
                  Computational Intelligence and Security (CIS), 2013,
                  December 14--15",
 publisher     = "IEEE",
 address       = "Leshan, China",
 year          = "2013",
 pages         = "798--802",
 doi           = "10.1109/CIS.2013.174",
 url           = "https://ieeexplore.ieee.org/document/6746542",
 annote        = "A set of matrices is said to have the finiteness property if
                  the maximal rate of growth of long products of matrices
                  taken from the set can be obtained by a periodic product.
                  We study the finite-step realizability of the
                  joint/generalized spectral radius of a pair of $n\times n$
                  square matrices. Let $\Sigma= \{A, B\}$ where $A, B$ are
                  $n\times n$ matrices and $B$ is a rank-one matrix. Then we
                  have $\rho(\Sigma)=
                  \max_{t,s}\rho(A^{t}B^{s})^\frac{1}{s+t}$. That is to say,
                  $\Sigma$ have the finiteness property where the maximum is
                  attained at $(t, s)$ with the optimal sequence $A^t B^s$.",
}

@ARTICLE{WangCheng:LMA16,
 author        = "Wang, Xingping and Cheng, Zhaolin",
 title         = "Infinite products of uniformly paracontracting matrices",
 journal       = "Linear Multilinear Algebra",
 fjournal      = "Linear and Multilinear Algebra",
 year          = "2016",
 volume        = "64",
 number        = "5",
 pages         = "856--862",
 issn          = "0308-1087",
 mrclass       = "65F10 (15A99 47H09)",
 mrnumber      = "3479385",
 zblnumber     = "1346.65018",
 zblreviewer   = "Edgar Pereira",
 doi           = "10.1080/03081087.2015.1063577",
 url           = "https://www.tandfonline.com/doi/full/10.1080/03081087.2015.1063577",
 annote        = "We define uniform paracontraction for an arbitrary set of
                  $n\times n$ matrices and show that an infinite product of
                  matrices drawn from a uniformly paracontracting set is
                  convergent. Moreover, if the uniformly paracontracting set
                  is finite and the matrices are drawn in a regulated way,
                  the infinite product is exponentially convergent.",
}

@ARTICLE{WRDV:IEEETAC16,
 author        = "Wang, Y. and Roohi, N. and Dullerud, G. E. and Viswanathan,
                  M.",
 authauthor    = "Wang, {\relax{}Yu} and Roohi, N. and Dullerud, G. and
                  Viswanathan, M.",
 title         = "Stability Analysis of Switched Linear Systems defined by
                  Regular Languages",
 journal       = "IEEE Trans. Autom. Control",
 fjournal      = "IEEE Transactions on Automatic Control",
 year          = "2016",
 volume        = "PP",
 number        = "99",
 pages         = "1--1",
 month         = aug,
 issn          = "0018-9286",
 mrclass       = "93D20 (68Q45 93C65)",
 mrnumber      = "3641474",
 mrreviewer    = "Silvio Capobianco",
 zblnumber     = "1366.93558",
 doi           = "10.1109/TAC.2016.2599930",
 url           = "https://ieeexplore.ieee.org/document/7542531",
 annote        = "In this work, we study the stability of an autonomous
                  discrete-time linear switched system whose switching
                  sequences are generated by a Muller automaton. This system
                  arises in various engineering problems such as distributed
                  communication and automotive engine control. The asymptotic
                  stability of this system, referred to as regular asymptotic
                  stability (RAS), generalizes two well-known definitions of
                  stability of autonomous discrete-time linear switched
                  systems, namely absolute asymptotic stability (AAS) and
                  shuffle asymptotic stability (SAS). We also extend these
                  stability definitions to robust versions. We show that
                  absolute asymptotic stability, robust absolute asymptotic
                  stability and robust shuffle asymptotic stability are
                  equivalent to exponential stability. In addition, by using
                  the Kronecker product, we prove that a robust regular
                  asymptotic stability problem is equivalent to the
                  conjunction of several robust absolute asymptotic stability
                  problems.",
}

@MASTERSTHESIS{WangYu:14,
 author        = "Wang, Yu",
 authauthor    = "Wang, {\relax{}Yu}",
 title         = "Stability of linear autonomous systems under regular
                  switching sequences",
 school        = "Mechanical Sci {\&} Engineering",
 address       = "University of Illinois at Urbana-Champaign",
 year          = "2014",
 pagetotal     = "35",
 url           = "https://www.ideals.illinois.edu/handle/2142/50614",
 annote        = "In this work, we discuss the stability of a discrete-time
                  linear autonomous system under regular switching sequences,
                  whose switching sequences are generated by a Muller
                  automaton. This system arises in various engineering
                  problems such as distributed communication and automotive
                  engine control. The asymptotic stability of this system,
                  referred to as regular asymptotic stability, generalizes
                  two well-known definitions of stability of autonomous
                  discrete-time linear switched systems, namely absolute
                  asymptotic stability (AAS) and shuffle asymptotic stability
                  (SAS). We also extend these stability definitions to robust
                  versions. We prove that absolute asymptotic stability,
                  robust absolute asymptotic stability and robust shuffle
                  asymptotic stability are equivalent to exponential
                  stability. In addition, by using the Kronecker product, we
                  prove that a robust regular asymptotic stability problem is
                  equivalent to the conjunction of several robust absolute
                  asymptotic stability problems.",
}

@INPROCEEDINGS{WRDV:CDC14,
 author        = "Wang, Yu and Roohi, Nima and Dullerud, Geir E. and
                  Viswanathan, Mahesh",
 authauthor    = "Wang, {\relax{}Yu} and Roohi, Nima and Dullerud, Geir and
                  Viswanathan, Mahesh",
 title         = "Stability of linear autonomous systems under regular
                  switching sequences",
 booktitle     = "Proceedings of the 53d IEEE Conference on Decision and
                  Control (CDC), 15-17 Dec.",
 publisher     = "IEEE",
 address       = "Los Angeles, California, USA",
 year          = "2014",
 pages         = "5445--5450",
 doi           = "10.1109/CDC.2014.7040240",
 url           = "https://ieeexplore.ieee.org/document/7040240",
 annote        = "In this work, we discuss the stability of a discrete-time
                  linear autonomous system under regular switching sequences,
                  whose switching sequences are generated by a Muller
                  automaton. This system arises in various engineering
                  problems such as distributed communication and automotive
                  engine control. The asymptotic stability of this system,
                  referred to as regular asymptotic stability, generalizes
                  two well-known definitions of stability of autonomous
                  discrete-time linear switched systems, namely absolute
                  asymptotic stability (AAS) and shuffle asymptotic stability
                  (SAS). We also extend these stability definitions to robust
                  versions. We prove that absolute asymptotic stability,
                  robust absolute asymptotic stability and robust shuffle
                  asymptotic stability are equivalent to exponential
                  stability. In addition, by using the Kronecker product, we
                  prove that a robust regular asymptotic stability problem is
                  equivalent to the conjunction of several robust absolute
                  asymptotic stability problems.",
}

@ARTICLE{WangJu:IEEECSL21,
 author        = "Wang, Zheming and Jungers, Rapha{\"e}l M.",
 authauthor    = "Zheming Wang and Rapha{\"e}l Jungers",
 title         = "A data-driven method for computing polyhedral invariant sets
                  of black-box switched linear systems",
 journal       = "IEEE Control Syst. Lett.",
 fjournal      = "IEEE Control Systems Letters",
 year          = "2021",
 volume        = "5",
 number        = "5",
 pages         = "1843--1848",
 eprinttype    = "arXiv",
 eprint        = "2009.10984",
 mrclass       = "93D05 (37 68 93C30)",
 mrnumber      = "4213953",
 doi           = "10.1109/LCSYS.2020.3044838",
 url           = "https://ieeexplore.ieee.org/document/9295324",
 annote        = "In this paper, we consider the problem of invariant set
                  computation for black-box switched linear systems using
                  merely a finite set of observations of system simulations.
                  In particular, this paper focuses on polyhedral invariant
                  sets. We propose a data-driven method based on the one step
                  forward reachable set. For formal verification of the
                  proposed method, we introduce the concept of
                  almost-invariant sets for switched linear systems. The
                  convexity-preserving property of switched linear systems
                  allows us to conduct contraction analysis on
                  almost-invariant sets and derive an a priori probabilistic
                  guarantee. In the spirit of non-convex scenario
                  optimization, we also establish a posteriori the level of
                  violation on the computed set. The performance of our
                  method is then illustrated by a switched system under
                  arbitrary switching between two modes.",
}

@INPROCEEDINGS{Wirth:CDC05,
 author        = "Wirth, F.",
 title         = "On the Structure of the Set of Extremal Norms of a Linear
                  Inclusion",
 booktitle     = "Proceedings of the 44th {IEEE} Conference on Decision and
                  Control, and the European Control Conference 2005 Seville,
                  Spain, December 12--15",
 year          = "2005",
 pages         = "3019--3024",
 doi           = "10.1109/CDC.2005.1582624",
 url           = "https://ieeexplore.ieee.org/document/1582624",
 annote        = "A systematic study of the set of extremal norms of an
                  irreducible linear inclusion is undertaken. We recall basic
                  methods for the construction of extremal norms, and
                  consider the action of basic operations from convex
                  analysis on these norms. It is shown that the set of
                  extremal norms of an irreducible linear inclusion is a
                  convex cone with a compact basis in an appropriate Banach
                  space. Furthermore, the compact basis may be chosen to
                  depend upper semi-continuously on the data. We explain that
                  this is the reason for the local Lipschitz continuity of
                  the joint spectral radius as a function of the data.",
}

@ARTICLE{Wirth:IJRNC98,
 author        = "Wirth, Fabian",
 title         = "On the calculation of time-varying stability radii",
 journal       = "Internat. J. Robust Nonlinear Control",
 fjournal      = "International Journal of Robust and Nonlinear Control",
 year          = "1998",
 volume        = "8",
 number        = "12",
 pages         = "1043--1058",
 coden         = "IJRCEA",
 issn          = "1049-8923",
 mrclass       = "93D09 (93B40)",
 mrnumber      = "1647729 (99g:93071)",
 zblnumber     = "0959.93044",
 zblreviewer   = "Mahesh G. Nerurkar (Camden)",
 doi           = "10/b8zb6x",
 url           = "https://onlinelibrary.wiley.com/doi/10.1002/(SICI)1099-1239(1998100)8:12\%3C1043::AID-RNC364\%3E3.0.CO;2-H",
 annote        = "The paper addresses the problem of computing the maximal
                  Lyapunov exponent of a discrete inclusion. This problem is
                  formulated as an ``average yield optimal control'' problem.
                  It is shown that the maximum value of this optimization
                  problem can be approximated by the maximal value of a
                  ``discounted optimal control problem''. This result is used
                  to obtain convergence rates of an algorithm for computing
                  the time-varying stability radii.",
}

@ARTICLE{Wirth:SIAMJCO98,
 author        = "Wirth, Fabian",
 title         = "Dynamics of time-varying discrete-time linear systems:
                  spectral theory and the projected system",
 journal       = "SIAM J. Control Optim.",
 fjournal      = "SIAM Journal on Control and Optimization",
 year          = "1998",
 volume        = "36",
 number        = "2",
 pages         = "447--487",
 issn          = "0363-0129",
 mrclass       = "93B50 (34D08 93C55)",
 mrnumber      = "1616502 (99i:93022)",
 mrreviewer    = "Russell A. Johnson",
 zblnumber     = "0942.93023",
 zblreviewer   = "M.G. Nerurkar",
 doi           = "10.1137/S0363012996299600",
 url           = "https://epubs.siam.org/doi/10.1137/S0363012996299600",
 annote        = "We study structural properties of linear time-varying
                  discrete-time systems. At first an associated system on
                  projective space is introduced as a basic tool to
                  understand the linear dynamics. We study controllability
                  properties of this system and characterize in particular
                  the control sets and their cores. Sufficient conditions for
                  an upper bound on the number of control sets with nonempty
                  interior are given. Furthermore, exponential growth rates
                  of the linear system are studied. Using finite-time
                  controllability properties in the cores of control sets the
                  Floquet spectrum of the linear system may be described. In
                  particular, the closure of the Floquet spectrum is
                  contained in the Lyapunov spectrum.\par This article lays a
                  foundation for the study of structural properties of linear
                  time-varying discrete time control processes. The basic
                  idea in this approach is to study the system on the
                  projective space (Bogolyubov's projection introdued by
                  Has'minskii). For deterministic, continuous-time systems
                  this program was carried out by F.~Colonius and
                  W.~Kliemann. In this paper the author studies asymptotic
                  properties of the system on the projective space, shows the
                  existence of controls with universal properties and
                  develops tools to analyze different dynamical notions of
                  `spectra'. The author studies controllability properties of
                  the system in terms of `control sets and their cores'.
                  Sufficient conditions for an upper bound on the number of
                  control sets with non-empty interior are given. Using
                  finite time controllability properties in the cores of
                  control sets, the Floquet spectrum of the linear system may
                  be described. In particular, the author shows that the
                  closure of the Floquet spectrum is contained in the
                  Lyapunov spectrum.",
}

@ARTICLE{Wirth:LAA02,
 author        = "Wirth, Fabian",
 title         = "The generalized spectral radius and extremal norms",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2002",
 volume        = "342",
 pages         = "17--40",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15A60 (34D08 93D09)",
 mrnumber      = "1873424 (2003g:15025)",
 mrreviewer    = "Zden{\v{e}}k Dost{\'a}l",
 zblnumber     = "0996.15020",
 zblreviewer   = "Alexey Alimov (Moskva)",
 doi           = "10.1016/S0024-3795(01)00446-3",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379501004463",
 annote        = "Two properties of the generalized spectral radius are
                  established, namely, its locally Lipschitz continuity on
                  the space of compact irreducible sets of matrices and its
                  strict monotonicity property. The author's approach is
                  based on an important idea in the analysis of exponential
                  stability of discrete inclusions that was introduced by
                  \emph{N.\,E.~Barabanov}~\cite{Bar:AIT88-2:e,Bar:AIT88-3:e,Bar:AIT88-5:e}.
                  The author gives a new proof for Barabanov's result, which
                  states that for irreducible sets of matrices an extremal
                  norm always exists. Also, significant conditions for the
                  existence of extremal norms are obtained.",
}

@INPROCEEDINGS{Wirth:MNTS02,
 author        = "Wirth, Fabian",
 title         = "Parameter Dependent Extremal Norms for Linear Parameter
                  Varying Systems",
 booktitle     = "Fifteenth International Symposium on Mathematical Theory of
                  Networks and Systems, August 12-16, 2002",
 organization  = "University of Notre Dame",
 year          = "2002",
 url           = "https://www3.nd.edu/~mtns/papers/9024.pdf",
 annote        = "We study families of time-varying linear systems with
                  restrictions on the derivative of the parameter variation.
                  This includes the systems usually considered in the area of
                  linear parameter varying (LPV) systems. We show that it is
                  possible to construct exact parameterized Lyapunov norms
                  for a wide class of such systems. This may be used to
                  derive (locally Lipschitz) continuous dependence of the
                  exponential growth rate on the systems data. Furthermore,
                  it is shown that the exponential growth rate may be
                  approximated by exponential growth rates of periodic
                  parameter variations.",
}

@MISC{Wirth:TR09,
 author        = "Wirth, Fabian",
 title         = "Further results on the {L}yapunov spectrum of time-varying
                  linear discrete-time systems",
 howpublished  = "Manuscript",
 year          = "2003",
 url           = "https://www.hamilton.ie/fabian/Archive/mtns-lyap.pdf",
 note          = "{Z}entrum f{\"u}r Technomathematik, Universit{\"a}t Bremen",
 annote        = "We continue our analysis of the Lyapunov and Floquet
                  spectrum of discrete-time systems begun
                  in~\cite{Wirth:SIAMJCO98}. In particular we settle an open
                  question by presenting an example of an invariant control
                  set in which Lyapunov exponents may be realized that are
                  not contained in the closure of the Floquet spectrum
                  corresponding to the control set. Furthermore we show that
                  under weak conditions the top Lyapunov exponents behaves
                  strictly monotone under monotone growth of the families of
                  time-varying systems. This implies a strong linearization
                  result for stability radii of nonlinear system with respect
                  to time-varying perturbations.",
}

@INPROCEEDINGS{Wirth:MTNS04,
 author        = "Wirth, Fabian",
 title         = "A monotonicity property of the joint spectral radius",
 booktitle     = "16th International Symposium on Mathematical theory of
                  networks and systems",
 publisher     = "Katholieke Universiteit Leuven",
 address       = "Heverlee",
 year          = "2004",
 url           = "https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.60.3589",
 annote        = "We show that the joint spectral radius of a set of matrices
                  is strictly increasing as a function of the data in the
                  sense that if a set of matrices is contained in the
                  relative interior of the convex hull of an irreducible set
                  of matrices, then the joint spectral radius of the smaller
                  set is strictly smaller than that of the larger set. This
                  observation has some consequences in the theory of
                  time-varying stability radii and their calculation. We show
                  by example that, strict monotonicity notwithstanding, $0$
                  may be a proximal normal of the joint spectral radius of
                  some (finitely parameterized) matrix polytopes functions.
                  This shows that the time-varying stability radius is not in
                  general Lipschitz continuous when it is continuous.",
}

@ARTICLE{Wirth:LAA05,
 author        = "Wirth, Fabian",
 title         = "The generalized spectral radius is strictly increasing",
 journal       = "Linear Algebra Appl.",
 fjournal      = "Linear Algebra and its Applications",
 year          = "2005",
 volume        = "395",
 pages         = "141--153",
 coden         = "LAAPAW",
 issn          = "0024-3795",
 mrclass       = "15A18 (34D08 39A10 93D09)",
 mrnumber      = "2112880 (2005m:15027)",
 mrreviewer    = "Nicola Guglielmi",
 zblnumber     = "1070.15007",
 zblreviewer   = "Jaspal Singh Aujla (Jalandhar)",
 doi           = "10.1016/j.laa.2004.07.013",
 url           = "https://www.sciencedirect.com/science/article/pii/S0024379504003374",
 annote        = "Given a nonempty compact set of matrices
                  $\mathcal{M}\subset\mathbb{K}^{n\times n}$, where
                  $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$. Consider the
                  discrete linear inclusion $x(t+1)\in\{Ax(t):A\in
                  \mathcal{M}\}$. A solution to this inclusion is a sequence
                  $\{x(t)\}_{t\in\mathbb{N}}$, such that for every
                  $t\in\mathbb{N}$ there is an $A(t)\in\mathcal{M}$ with
                  $x(t+1)=A(t)x(t)$. The sets of products of length $t$ are
                  defined as: $\mathcal{S}_t =\{A(t-1)A(t-2)\cdots
                  A(0):A(s)\in\mathcal{M}$, $s=0,1,\ldots ,t-1\}$ and the
                  semigroup given by $\mathcal{S}=
                  \bigcup_{t=0}^{\infty}\mathcal{S}_t$.\par Let $\rho(A)$
                  denote the spectral radius of $A$ and let $\|\cdot\|$ be
                  some operator norm on $ \mathbb{K}^{n\times n}$. Define for
                  $t\in\mathbb{N}$, $\bar{\rho}_t(\mathcal{M}):=
                  \sup\{r(S_t)^{1/t} : S_t \in\mathcal{S}_t\}$ and
                  $\hat{\rho}_t(\mathcal{M}):= \sup\{\|S_t\|^{1/t} :S_t
                  \in\mathcal{S}_t\}$ Then the joint spectral radius and the
                  generalized spectral radius are respectively defined as
                  $\bar{\rho}(\mathcal{M}):=\lim_{t\rightarrow\infty}
                  \sup\bar{\rho}_t(\mathcal{M})$ and
                  $\hat{\rho}(\mathcal{M}):=\lim_{t\rightarrow
                  \infty}\hat{\rho}_t(\mathcal{M})$. It is well known that
                  $\bar{\rho}(\mathcal{M})=\hat{\rho}(\mathcal{M})$.
                  $\mathcal{M}$ is called irreducible, if only the trivial
                  subspaces $\{0\}$ and $\mathbb{K}^n$ are invariant under
                  all matrices $A\in\mathcal{M}$, otherwise $\mathcal{M}$ is
                  called reducible. The semigroup $\mathcal{S}$ is
                  irreducible if and only if the set $\mathcal{M}$ is
                  irreducible.\par The author proves that the generalized
                  spectral radius of a compact set of matrices is a strictly
                  increasing function of the set. The main tool in the proof
                  is the observation that if $\mathcal{M}$ is convex, is not
                  a singleton set, and the semigroup $\mathcal{S}$ generated
                  by $\mathcal{M}$ satisfies $\sigma(S)\subset\{0\}\cup
                  \{z\in\mathbb{C}:|z| =1\},\forall
                  S\in\mathcal{S}(\mathcal{M})$, then $\mathcal{M}$ is
                  reducible.\par Some applications of this property in the
                  area of time-varying stability radii are discussed. In
                  particular, using the implicit function theorem sufficient
                  conditions for Lipschitz continuity are derived.",
}

@INPROCEEDINGS{Wirth:IFACWC05,
 author        = "Wirth, Fabian",
 title         = "The {G}elfand formula for linear parameter-varying and
                  linear switching systems",
 booktitle     = "Proceedings of the 16th IFAC World Congress",
 organization  = "IFAC",
 address       = "Czech Republic",
 year          = "2005",
 volume        = "16, part 1",
 pages         = "652--652",
 doi           = "10.3182/20050703-6-CZ-1902.00653",
 url           = "https://www.sciencedirect.com/science/article/pii/S1474667016366654",
 annote        = "It is shown that the Gelfand formula holds for a large class
                  of families of linear time-varying systems, encompassing in
                  particular standard formulations of linear
                  parameter-varying and linear switching systems. By this
                  result, the uniform exponential growth rate may be
                  approximated to arbitrary precision by the growth rate of
                  periodic systems within the family. This result extends
                  classical results in the area of linear inclusions. The
                  basic tool in the proof is a recent construction of
                  parameter dependent Lyapunov functions for the family of
                  linear time-varying systems that exactly characterize the
                  exponential growth rate.",
}

@ARTICLE{Wolf:PAMS,
 author        = "Wolfowitz, J.",
 title         = "Products of indecomposable, aperiodic, stochastic matrices",
 journal       = "Proc. Amer. Math. Soc.",
 fjournal      = "Proceedings of the American Mathematical Society",
 year          = "1963",
 volume        = "14",
 pages         = "733--737",
 issn          = "0002-9939",
 mrclass       = "15.65",
 mrnumber      = "0154756 (27 \#4702)",
 mrreviewer    = "J. Kiefer",
 zblnumber     = "0116.35001",
 doi           = "10.1090/S0002-9939-1963-0154756-3",
 url           = "https://www.ams.org/journals/proc/1963-014-05/S0002-9939-1963-0154756-3/",
 annote        = "A finite square matrix $P=\{p_{i j}\}$ is called stochastic
                  if $p_{i j}\ge0$ for all $i, j$, and $\sum_{j} p_{i j}=1$
                  for all $i$. A stochastic matrix $P$ is called
                  indecomposable and aperiodic (SIA) if $Q=\lim_{n\to\infty}
                  P^{n}$ exists and all the rows of $Q$ are the same. SIA
                  matrices are defined differently in books on probability
                  theory. \par We define $\delta(P)$ by $\delta(P) = \max_{j}
                  \max_{i_{1},i_{2}}|p_{i_{l}j} - p_{i_{2}j}|$. Thus
                  $\delta(P)$ measures, in a certain sense, how different the
                  rows of $P$ are. If the rows of $P$ are identical,
                  $\delta(P) =0$ and conversely. \par Let $A_{1}, \ldots,
                  A_{k}$ be any square matrices of the same order. By a word
                  (in the $A$'s) of length $t$ we mean the product of $t$
                  $A$'s (repetitions permitted). The object of this note is
                  to prove the following assertion:\par \textbf{Theorem.}
                  \emph{Let $A_{1}, \ldots, A_{k}$ be square stochastic
                  matrices of the same order such that any word in the $A$'s
                  is SIA. For any $\varepsilon>0$ there exists an integer
                  $\nu(e)$ such that any word $B$ (in the $A$'s) of length
                  $n\ge\nu(e)$ satisfies $\delta(B) < \varepsilon$.}\par In
                  words, the result is that any sufficiently long word in the
                  $A$'s has all its rows approximately the same.",
}

@ARTICLE{WuHe:SIAMJCO20,
 author        = "Wu, Zeyang and He, Qie",
 title         = "Optimal switching sequence for switched linear systems",
 journal       = "SIAM J. Control Optim.",
 fjournal      = "SIAM Journal on Control and Optimization",
 year          = "2020",
 volume        = "58",
 number        = "2",
 pages         = "1183--1206",
 issn          = "0363-0129",
 mrclass       = "90C27 (05A16 37N40 68Q25 90C40 93C30)",
 mrnumber      = "4091167",
 zblnumber     = "07203554",
 doi           = "10.1137/18M1197928",
 url           = "https://epubs.siam.org/doi/10.1137/18M1197928",
 annote        = "We study the following optimization problem over a dynamical
                  system that consists of several linear subsystems: Given a
                  finite set of $n\times n$ matrices and an $n$-dimensional
                  vector, find a sequence of $K$ matrices, each chosen from
                  the given set of matrices, to maximize a convex function
                  over the product of the $K$ matrices and the given vector.
                  This simple problem has many applications in operations
                  research and control, yet a moderate-sized instance is
                  challenging to solve to optimality for state-of-the-art
                  optimization software. We propose a simple exact algorithm
                  for this problem. Our algorithm runs in polynomial time
                  when the given set of matrices has the \emph{oligo-vertex}
                  property, a concept we introduce in this paper for a set of
                  matrices. We derive several sufficient conditions for a set
                  of matrices to have the oligo-vertex property. Numerical
                  results demonstrate the clear advantage of our algorithm in
                  solving large-sized instances of the problem over one
                  state-of-the-art global solver. We also pose several open
                  questions on the oligo-vertex property and discuss its
                  potential connection with the finiteness property of a set
                  of matrices, which may be of independent interest.",
}

@ARTICLE{Xu:EJLA10,
 author        = "Xu, Jianhong",
 title         = "On the trace characterization of the joint spectral radius",
 journal       = "Electron. J. Linear Algebra",
 fjournal      = "Electronic Journal of Linear Algebra",
 year          = "2010",
 volume        = "20",
 pages         = "367--375",
 issn          = "1081-3810",
 mrclass       = "15A18 (15A42 15A60)",
 mrnumber      = "2679735 (2011g:15022)",
 mrreviewer    = "Jaspal Singh Aujla",
 zblnumber     = "1217.15016",
 doi           = "10.13001/1081-3810.1381",
 url           = "https://journals.uwyo.edu/index.php/ela/article/view/763",
 annote        = "A characterization of the joint spectral radius, due to
                  \emph{Q.~Chen} and \emph{X.~Zhou}~\cite{ChenZhou:LAA00},
                  relies on the limit superior of the $k$-th root of the
                  dominant trace over products of matrices of length $k$. In
                  this note, a sufficient condition is given such that the
                  limit superior takes the form of a limit. This result is
                  useful while estimating the joint spectral radius. Although
                  it applies to a restricted class of matrices, it appears to
                  be relevant to many realistic situations.",
}

@ARTICLE{XuXiao:AJIFAC11,
 author        = "Xu, Jianhong and Xiao, Mingqing",
 title         = "A characterization of the generalized spectral radius with
                  {K}ronecker powers",
 journal       = "Automatica J. IFAC",
 fjournal      = "Automatica. A Journal of IFAC, the International Federation
                  of Automatic Control",
 year          = "2011",
 volume        = "47",
 number        = "7",
 pages         = "1530--1533",
 coden         = "ATCAA9",
 issn          = "0005-1098",
 mrclass       = "15A60",
 mrnumber      = "2889254 (2012m:15018)",
 mrreviewer    = "Hiroshi Nakazato",
 zblnumber     = "1227.15013",
 zblreviewer   = "Mihail Voicu (Ia{\c{s}}i)",
 doi           = "10.1016/j.automatica.2011.04.007",
 url           = "https://www.sciencedirect.com/science/article/pii/S0005109811002275",
 annote        = "Based on Tur{\'a}n's power sum theory, in this note a result
                  presented by \emph{V.\,D.~Blondel} and
                  \emph{Y.~Nesterov}~\cite{BN:SIAMJMA05} is extended in the
                  following form:\par \textbf{Theorem.} \emph{The generalized
                  spectral radius of $\Sigma=\{A_1,\ldots,A_m\}\subset
                  \mathbb{C}^{n\times n}$ is characterized by
                  $\rho(\Sigma)=\limsup_{k\to\infty}
                  \rho^{1/k}(A_{1}^{\otimes k} +\dots +A_{m}^{\otimes k})$,
                  where $A_{i}^{\otimes k}$ is the $k$th Kronecker power of
                  $A_{i}$.}",
}

@ARTICLE{XuAci:IEEETAC20,
 author        = "Xu, Xiangru and A{\c{c}\i}kme{\c{s}}e, Beh{\c{c}}et",
 title         = "Approximation of the Constrained Joint Spectral Radius via
                  Algebraic Lifting",
 journal       = "IEEE Trans. Automat. Control",
 fjournal      = "Institute of Electrical and Electronics Engineers.
                  Transactions on Automatic Control",
 year          = "2021",
 volume        = "66",
 number        = "7",
 pages         = "3386--3392",
 eprinttype    = "arXiv",
 eprint        = "1807.03965",
 issn          = "0018-9286",
 mrclass       = "93D20 (15A18 93B25 93C30)",
 mrnumber      = "4284164",
 zblnumber     = "1467.93165",
 doi           = "10.1109/TAC.2020.3020580",
 url           = "https://ieeexplore.ieee.org/document/9184221",
 annote        = "A linear constrained switching system is a discrete-time
                  linear switched system whose switching sequences are
                  constrained by a deterministic finite automaton. As a
                  characterization of the asymptotic stability of a
                  constrained switching system, the constrained joint
                  spectral radius is difficult to compute or approximate.
                  Using the semi-tensor product of matrices, we express
                  dynamics of a deterministic finite automaton, an arbitrary
                  switching system and a constrained switching system into
                  their matrix forms, respectively, where the matrix
                  expression of a constrained switching system can be seen as
                  the matrix expression of a lifted arbitrary switching
                  system. Inspired by this, we propose a lifting method for
                  the constrained switching system, and prove that the
                  constrained joint/generalized spectral radius of the
                  constrained switching system is equivalent to the
                  joint/generalized spectral radius of the lifted arbitrary
                  switching system. Examples are provided to show the
                  advantages of the proposed lifting method.",
}

@INCOLLECTION{Yoc:MDSA04,
 author        = "Yoccoz, Jean-Christophe",
 title         = "Some questions and remarks about {${\rm SL}(2,\mathbf{R})$}
                  cocycles",
 booktitle     = "Modern dynamical systems and applications",
 publisher     = "Cambridge Univ. Press",
 address       = "Cambridge",
 year          = "2004",
 pages         = "447--458",
 mrclass       = "37D20 (37B55 37C15)",
 mrnumber      = "2093316 (2005h:37068)",
 mrreviewer    = "Russell A. Johnson",
 zblnumber     = "1148.37306",
 url           = "https://www.college-de-france.fr/media/jean-christophe-yoccoz/UPL27219_SL_2_R__cocycles.pdf",
 annote        = "There have been many deep results about cocycle maps in
                  recent years, especially in the quasiperiodic case with the
                  achievements of \emph{H.~Eliasson}, \emph{J.~Bourgain} and
                  \emph{R.~Krikorian} amongst others. The questions that we
                  address here are much more elementary: most of the time, we
                  will be interested in locally constant cocycles with values
                  in $\textit{SL}(2,\mathbf{R})$ over a transitive subshift
                  of finite type; we want to determine whether this cocycle
                  map is uniformly hyperbolic and how it can bifurcate from
                  uniform hyperbolicity. In our setting, parameter space is
                  finite-dimensional, and we would like to describe it in the
                  same way that one does for polynomials or rational maps,
                  where hyperbolic components play a leading role in the
                  picture. It appears that even in this very much simplified
                  situation, several interesting questions appear.",
}

@INPROCEEDINGS{ZSD:CCC21,
 title         = "Semi-tensor Product Approach to Approximation of the
                  Constrained Joint Spectral Radius",
 author        = "Zhang, Haoyue and Song, Yang and Deng, Li",
 year          = "2021",
 booktitle     = "2021 40th Chinese Control Conference (CCC)",
 publisher     = "IEEE",
 pages         = "206--211",
 url           = "https://ieeexplore.ieee.org/abstract/document/9550754",
 doi           = "10.23919/CCC52363.2021.9550754",
 annote        = "In this paper, a semi-tensor product (STP) approach is
                  proposed to discuss the stability problem of a constrained
                  switching system (CSS). The switching sequence is subject
                  to the constraint of a nondeterministic finite automaton
                  (NFA). By the mathematical tool, the constrained joint
                  spectral radius (CJSR) of the CSS can be approximated. We
                  propose the equivalence of a matrix-based expression of the
                  CSS and its lifted l-dimensional switching system by using
                  the STP approach. After that, the approximate relation
                  between the constrained joint spectral radius (CJSR)/the
                  constrained generalized spectral radius (CGSR) and the
                  joint spectral radius (JSR)/generalized spectral radius
                  (GSR) is developed for the switching system. At the same
                  time, there is a special case of NFA, where the equivalence
                  relation between them also is proved.",
}

@MISC{ZhangXu:ArXiv20,
 author        = "Zhang, Yuhao and Xu, Xiangru",
 title         = "Finding matrix sequences with a high asymptotic growth rate
                  for linear constrained switching systems",
 howpublished  = "ArXiv.org e-Print archive",
 year          = "2020",
 month         = sep,
 eprinttype    = "arXiv",
 eprint        = "2009.12948",
 doi           = "10.48550/arXiv.2009.12948",
 annote        = "Linear constrained switching systems are linear switched
                  systems whose switching sequences are constrained by a
                  deterministic finite automaton. This work investigates how
                  to generate a sequence of matrices with an asymptotic
                  growth rate close to the constrained joint spectral radius
                  (CJSR) for constrained switching systems, based on our
                  previous result that reveals the equivalence of a
                  constrained switching system and a lifted arbitrary
                  switching system. By using the dual solution of a
                  sum-of-squares optimization program, an algorithm is
                  designed for the lifted arbitrary switching system to
                  produce a sequence of matrices with an asymptotic growth
                  rate that is close to the CJSR of the original constrained
                  switching system. It is also shown that a type of existing
                  algorithms designed for arbitrary switching systems can be
                  applied to the lifted system such that the desired sequence
                  of matrices can be generated for the constrained switching
                  system. Several numerical examples are provided to
                  illustrate the better performance of the proposed
                  algorithms compared with existing ones.",
}

@ARTICLE{Zhou:AMC06,
 author        = "Zhou, Xinlong",
 title         = "Estimates for the joint spectral radius",
 journal       = "Appl. Math. Comput.",
 fjournal      = "Applied Mathematics and Computation",
 year          = "2006",
 volume        = "172",
 number        = "1",
 pages         = "332--348",
 coden         = "AMHCBQ",
 issn          = "0096-3003",
 mrclass       = "15A18",
 mrnumber      = "2197905",
 zblnumber     = "1104.65033",
 zblreviewer   = "Ronald H. W. Hoppe",
 doi           = "10.1016/j.amc.2005.02.006",
 url           = "https://www.sciencedirect.com/science/article/pii/S0096300305002067",
 annote        = "Estimates for the computation of the joint spectral radius
                  defined by a bounded collection of square matrices and of
                  same size will be given. In particular, the computational
                  result shows that for matrices constructed by
                  four-coefficient two-scale dilation equation our approach
                  leads to the exact value of this joint spectral radius.",
}

@ARTICLE{ZCL:DCDSA18,
 author        = "Zou, Rui and Cao, Yongluo and Liao, Gang",
 title         = "Continuity of spectral radius over hyperbolic systems",
 journal       = "Discrete Contin. Dyn. Syst.",
 fjournal      = "Discrete and Continuous Dynamical Systems. Series A. A
                  Journal Bridging Mathematics and Sciences",
 year          = "2018",
 volume        = "38",
 number        = "8",
 pages         = "3977--3991",
 issn          = "1078-0947",
 mrclass       = "37H15 (15A60 34D08 37C50 37D25)",
 mrnumber      = "3814361",
 mrreviewer    = "Miaohua Jiang",
 zblnumber     = "1396.37061",
 doi           = "10.3934/dcds.2018173",
 url           = "https://www.aimsciences.org/article/doi/10.3934/dcds.2018173",
 annote        = "The continuity of joint and generalized spectral radius is
                  proved for H{\"o}lder continuous cocycles over hyperbolic
                  systems. We also prove the periodic approximation of
                  Lyapunov exponents for non-invertible non-uniformly
                  hyperbolic systems, and establish the Berger--Wang formula
                  for general dynamical systems.",
}