J. Nonsmooth Anal. Optim. 1 (2020), 5800, doi: 10.46298/jnsao-2020-5800
Submitted: 2019-10-01, accepted: 2020-03-19

page 1 of 41
© the authors, cc-by-sa 4.0

optimal control of an abstract evolution
variational inequality with application to
homogenized plasticity
Hannes Meinlschmidt∗

Christian Meyer†

Stephan Walther†

Abstract The paper is concerned with an optimal control problem governed by a state equation
in form of a generalized abstract operator differential equation involving a maximal monotone
operator. The state equation is uniquely solvable, but the associated solution operator is in general
not Gâteaux differentiable. In order to derive optimality conditions, we therefore regularize the state
equation and its solution operator, respectively, by means of a (smoothed) Yosida approximation.
We show convergence of global minimizers for regularization parameter tending to zero and derive
necessary and sufficient optimality conditions for the regularized problems. The paper ends with an
application of the abstract theory to optimal control of homogenized quasi-static elastoplasticity.

1 introduction
This paper is concerned with an optimal control problem of the following form, governed by an operator
differential equation:

(P)











min J (z, `) B Ψ(z, `) + Φ(`),
.

s.t. z ∈ A(R` − Qz),

z(0) = z 0 ,



(z, `) ∈ H (0,T ; H ) × H 1 (0,T ; Xc ) ∩ U(z 0 ; M) .
1

Herein, A is a maximal monotone operator, while R and Q are linear and bounded operators in a
Hilbert space H . The control variable is denoted by `, whereas z is the state of the system. The precise
assumptions on the data are given in Section 2 below.
The particular feature of the problem under consideration is the set-valued mapping A. Due to its
maximal monotony, one can show that there is a well-defined single-valued control-to-state mapping
` 7→ z (in suitable function spaces), but this mapping is in general not Gâteaux differentiable. We are
thus faced with a non-smooth optimal control problem, for which the derivation of necessary and
sufficient optimality conditions is a particular challenge.
Depending on the precise choice of A, R, and Q, problem (P) covers various application problems.
For instance, quasi-static elastoplasticity is frequently modeled in this way. Here, R is the solution
operator associated with the equations of linear elasticity for given load distribution `. Moreover,
This research was supported by the German Research Foundation (DFG) under grant number ME 3281/9-1 within the priority
program Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization
(SPP 1962).
∗ Johann Radon Institute for Computational and Applied Mathematics (RICAM), Alternberger Straße 66a, 4040 Linz, Austria

(hannes.meinlschmidt@ricam.oeaw.ac.at, https://people.ricam.oeaw.ac.at/h.meinlschmidt/)

† TU Dortmund, Faculty of Mathematics, Vogelpothsweg 87, 44227 Dortmund, Germany (christian2.meyer@tu-dortmund.de,

http://www.mathematik.tu-dortmund.de/lsx/cms/de/mitarbeiter/cmeyer.html, stephan.walther@tu-dortmund.de, http:
//www.mathematik.tu-dortmund.de/lsx/cms/de/mitarbeiter/swalther.html).

J. Nonsmooth Anal. Optim. 1 (2020), 5800

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Q is the sum of the solution operator of linear elasticity for given stress distribution, the elasticity
tensor, and a coercive operator modeling hardening effects. Finally, A is the convex subdifferential of
the indicator functional of the closed and convex set of feasible stresses defined by a suitable yield
condition. Details on models in elastoplasticity can be found in [25]. Another model which is covered
by the state equation in (P) is the system of homogenized elastoplasticity, which we will study in detail
in Section 7 below.
Let us put our work into perspective. Assume for a moment that A is the convex subdifferential of a
proper, convex, and lower semicontinuous functional ϕ and that Q is self adjoint. Then, by convex
duality, the state equation is equivalent to
.

(1.1)

0 ∈ ∂ϕ ∗ (z) + E 0(z),

z(0) = z 0 ,

where E is the quadratic energy functional given by
E(z) B 21 (Qz, z) H − hR`, zi.
Systems of this type have been intensively studied concerning existence of solutions and their numerical
approximation, and we only refer to [29] and the references therein. In contrast to this, the literature on
optimization problems governed by (1.1) is rather scarce. The research on optimal control of equations
of type (1.1) probably started with the sweeping process, where ϕ = I −C(t ) is the indicator functional
of a moving convex set C(t), see [30]. In the optimal control setting, C(t) is most frequently set
to C(t) = `(t) − Z with a convex set Z and a driving force `. This fits into the setting of (1.1) by
defining ϕ B I Z and Q = R = id (identity). Optimal control problems of this type are investigated
in [1, 2, 9–11, 15–17], where the underlying Hilbert space is mostly finite dimensional. Problems in an
infinite dimensional Hilbert space are investigated in [21,37]. To be more precise, in these contributions,
H is the Sobolev space H 01 (Ω) and E is the Dirichlet energy. Moreover, ϕ ∗ is set to ϕ ∗ (z) = kzkL1 (Ω) and
its viscous regularization, respectively, i.e., ϕ δ∗ (z) = kzkL1 (Ω) + δ2 kzkH2 1 (Ω) . Optimal control problems
0

governed by quasi-static elastoplasticity with linear kinematic hardening and von Mises yield conditions
are treated in [40, 42, 43]. As already indicated above, ϕ is the indicator functional of the convex set of
feasible stresses in this case. All mentioned problems fit into our framework and can thus be seen as
special cases of our non-smooth evolution. Our analysis therefore represents a generalization of existing
results on optimal control of non-smooth evolution problems and can also be applied to application
problems that were not treated in the literature so far such as optimal control of homogenized plasticity,
which is investigated in Section 7. We emphasize that problems with non-convex energies such as
damage evolution are not covered by our analysis. Optimal control problems governed by (1.1) with
non-convex energy are investigated in [31, 32].
Our strategy to analyze (P) is as follows: After showing well-posedness of the state equation and
the optimal control problem, we employ the Yosida-regularization with an additional smoothing to
obtain a smooth (i.e., Fréchet differentiable) control-to-state map. We will prove that accumulation
points of global minimizers of the regularized optimal control problems for vanishing regularization
are solutions of the original non-smooth problem (P). Moreover, first-order necessary and second-order
sufficient optimality conditions for the regularized problems are derived. The passage to the limit to
establish optimality conditions for the original problem goes beyond the scope of this paper and is
subject to future work. The results of [37, 43] indicate that the optimality conditions obtained in this
way are rather weak and we expect the same all the more for our general setting. Let us underline that
regularization is a widely used approach to treat optimal control problems governed by non-smooth
evolutions. We only refer to [10, 21, 42] and the references therein.
The paper is organized as follows. After stating our standing assumptions in Section 2, we investigate
the state equation and its regularization in Section 3. In Section 4, we then turn to the optimal control
problem and show that it admits an optimal solution under our standing assumptions. Moreover, we
H. Meinlschmidt, C. Meyer, S. Walther

Optimal control of an abstract evolution variational . . .

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establish the convergence result for vanishing regularization indicated above. Section 5 is devoted
to first-order necessary optimality conditions for the regularized problems in form of a KKT-system
involving an adjoint equation. In Section 6, we address second-order sufficient conditions for the
regularized problem. The paper ends with the adaptation of our general results to a concrete application
problem, namely the optimal control of homogenized elastoplasticity.

2 notation and standing assumptions
We start with a short introduction in the notation used throughout the paper.
Notation Given two vector spaces X and Y, we denote the space of linear and continuous functions
from X into Y by L(X, Y). If X = Y, we simply write L(X). The dual space of X is denoted by
X ∗ = L(X, R). If H is a Hilbert space, we denote its inner product by (·, ·) H . For the whole paper, we
fix the final time T > 0. We denote the Bochner space of square-integrable functions by L2 (0,T ; X)
and the Bochner-Sobolev space by H 1 (0,T ; X). Given a coercive operator G : H → H in a Hilbert
2 for all h ∈ H . Finally, c > 0
space H , we denote its coercivity constant by γG , i.e., (Gh, h) H ≥ γG khk H
and C > 0 denote generic constants.

standing assumptions
The following standing assumptions are tacitly assumed for the rest of the paper without mentioning
them every time.
Spaces Throughout the paper, X, Xc , Y, Z, W are real Banach spaces. Moreover, Xc reflexive and
H is a separable Hilbert space. The space Xc is compactly embedded into X and the embeddings
Y ,→ Z ,→ H ,→ W are continuous.
Operators
(2.1)

The operator A: H → 2 H is maximal monotone, its domain D(A) is closed and we define
A0 : D(A) → H,

h 7→ arg min kv k H .
v ∈A(h)

Furthermore, by Aλ : H → H , λ > 0, we denote the Yosida-approximation of A and by R λ = (I + λA)−1
the resolvent of A, so that Aλ = λ1 (I − R λ ). We assume that the operator A0 : D(A) → H is bounded on
bounded sets. For further reference on maximal monotone operators, we refer to [7], [44, Ch. 32], [45,
Ch. 55], and [35, Ch. 55]. Furthermore, R ∈ L(X; Y) and Q ∈ L(W; W), are linear and bounded
operators, and the restriction of Q to H , Z, or Y maps into these spaces and is again linear and
bounded. To ease notation, we denote this restriction by the same symbol. Moreover, Q : H → H is
coercive and self-adjoint.
Optimization Problem By J : H 1 (0,T ; W) × H 1 (0,T ; Xc ) → R we denote the objective function. We
assume that both Ψ : H 1 (0,T ; W) × H 1 (0,T ; Xc ) → R and Φ : H 1 (0,T ; Xc ) → R are weakly lower
semicontinuous. Moreover, Ψ is bounded from below and continuous in the first argument, while Φ is
coercive. The set M is a nonempty and closed subset of D(A) and z 0 ∈ Y is a given initial state.
Remark 2.1. We emphasize that not all of the above assumptions are always needed. For instance, in the
next two sections, Q and R are only considered as operators with values in H and the spaces Y and Z
are not needed, before the investigation of optimality conditions starts in Sections 5 and 6. However,
to keep the discussion concise, we present the standing assumption in the present form.
H. Meinlschmidt, C. Meyer, S. Walther

Optimal control of an abstract evolution variational . . .

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3 state equation
3.1 existence and uniqueness
We start the investigation of (P) with the discussion of the state equation, i.e.,
.

z ∈ A(R` − Qz),

(3.1)

z(0) = z 0 .

Definition 3.1. Let ` ∈ H 1 (0,T ; X) and z 0 ∈ H . Then z ∈ H 1 (0,T ; H ) is called solution of (3.1), if
.
z(0) = z 0 and z(t) ∈ A(Rl(t) − Qz(t)) holds for almost all t ∈ [0,T ].
In order to obtain the existence of a solution to (3.1), the data have to fulfill a certain compatibility
condition. For this reason, we introduce the following
Definition 3.2. For z 0 ∈ H and M ⊂ D(A), we define the set

U(z 0 , M) B ` ∈ H 1 (0,T ; X) : R`(0) − Qz 0 ∈ M
of admissible loads.
Theorem 3.3 (Existence result for the state equation). Let z 0 ∈ H and ` ∈ U(z 0 , D(A)). Then there
exists a unique solution z ∈ H 1 (0,T ; H ) of (3.1). Furthermore, there exists a constant C, independent of
z 0 and `, such that
.


(3.2)
kz kC([0,T ];H) ≤ C 1 + kz 0 k H + k`kC([0,T ];X) + k `kL1 (0,T ;X) ,
.


.
(3.3)
kzkL2 (0,T ;H) ≤ C k `kL2 (0,T ;X) + sup kA0 (R`(τ ) − Qz(τ ))k H ,
τ ∈[0,T ]

where A0 is as defined in (2.1).
Proof. The proof essentially follows the lines of [24, Theorem 4.1] and [7, Proposition 3.4]. For convenience of the reader, we sketch the main arguments. At first, one employs the transformation
H 1 (0,T ; H ) 3 z 7→ q B R` − Qz ∈ H 1 (0,T ; H ) with its inverse H 1 (0,T ; H ) 3 q 7→ z B Q −1 (R` − q) ∈
H 1 (0,T ; H ) to see that (3.1) is equivalent to
.

.

(3.4)

q + QA(q) 3 R `,

q(0) = R`(0) − Qz 0 .

Since Q is coercive the operator, Ã: H → 2 H , h 7→ QA(h) is maximal monotone with respect to the
scalar product


(h 1 , h 2 ) H,Q −1 B Q −1h 1 , h 2 H , h 1 , h 2 ∈ H .
Therefore, [7, Proposition 3.4] yields the existence of a unique solution q ∈ H 1 (0,T ; H ) of (3.4). To
verify the estimate in (3.2), we employ [7, Lemme 3.1], which gives
ˆ t .
kq(t) − q̃(t)k H,Q −1 ≤ kR`(0) − Qz 0 − ak H,Q −1 +
kR `(τ )k H,Q −1 dτ ,
0

where q̃ is the unique solution of
.

q̃ + QA(q̃) 3 0,

q(0) = a

with an arbitrary element a ∈ D(A). This gives the desired first inequality.
To prove the second inequality, we deduce from [7, Proposition 3.4] and the associated proof that
ˆ t .
kq(t) − q(s)k H,Q −1 ≤
kR `(τ )k H,Q −1 dτ + sup kÃ0 (q(τ ))k H,Q −1 (t − s).
s

H. Meinlschmidt, C. Meyer, S. Walther

τ ∈[0,T ]

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Dividing this inequality by (t − s) and letting t → s yields
.

.

(3.5)

kq(s)k H,Q −1 ≤ kR `(s)k H,Q −1 + sup kÃ0 (q(τ ))k H,Q −1
τ ∈[0,T ]

for almost all s ∈ [0,T ]. From the definition of Ã0 (with respect to (·, ·) H,Q −1 ) we see that kÃ0 (h)k H,Q −1 ≤
kQv k H,Q −1 for all v ∈ A(h). This holds in particular for v = A0 (h) so that z = Q −1 (R` − q) and (3.5)
imply
 .

.


.
.
kz(s)k H ≤ C k `(s)k X + kq(s)k H,Q −1 ≤ C k `(s)k X + sup kA0 (R`(τ ) − Qz(τ ))k H ,
τ ∈[0,T ]



which gives the second inequality.

Remark 3.4. In order to prove Theorem 3.3, it is sufficient to require that A0 is bounded on compact
subsets (in addition to the closedness of D(A)), cf. [7, Proposition 3.4]. However, the boundedness on
bounded sets of A0 is needed to prove Theorem 3.11 below and therefore, we impose it as a standing
assumption.
Based on Theorem 3.3, we may introduce the solution operator associated with (3.1) and reduce (P)
to an optimization problem in the control variable only, see Definition 4.1 below. Due to the set-valued
operator A, this solution operator will in general be non-smooth, which complicates the derivation of
first- and second-order optimality conditions. A prominent way to overcome this issue is to regularize
the state equation in order to obtain a smooth solution operator. This is frequently done by means of
the Yosida-approximation, see e.g. [5], and we will pursue the same approach. For this purpose, we
will investigate the Yosida-approximation and its convergence properties in the next subsection.

3.2 regularization and convergence results
For the rest of this section, we fix z 0 ∈ H and ` ∈ U(z 0 , D(A)) and denote the unique solution of (3.1)
by z. We start with a convergence result of the Yosida-approximation for fixed data z 0 and ` and then
turn to perturbation of the data.
Proposition 3.5 (Convergence of the Yosida-approximation for fixed data). Let z λ ∈ H 1 (0,T ; H ) be the
solution of
.

z λ = Aλ (R` − Qz λ ),

(3.6)

z λ (0) = z 0

for all λ > 0. Then z λ → z in H 1 (0,T ; H ) as λ & 0 and the following inequality holds
(3.7)

2
kz λ − z kC([0,T
];H) +

λ . 2
λ .
λ . 2
.
kz λ kL2 (0,T ;H) +
kz λ − zkL2 2 (0,T ;H) ≤
kzkL2 (0,T ;H) .
γQ
γQ
γQ

Proof. The proof in principle follows the lines of [7, Proposition 3.11], since our assumptions and
assertions however are slightly different, we provide the arguments in detail.
First of all, since z 7→ Aλ (R` − Qz) is Lipschitz-continuous by [45, Proposition 55.2(b)], the existence
of a unique solution of (3.6) follows from Banach’s contraction principle by standard arguments,
cf. e.g. [19]. Moreover, [45, Proposition 55.2(a)] and the definition of Aλ give


d
.
.
(Q(z λ (t) − z(t)), z λ (t) − z(t)) H = 2 z λ (t) − z(t), Q(z λ (t) − z(t)) H
dt

 


.
.
= −2 z λ (t) − z(t), R λ R`(t) − Qz λ (t) − R`(t) − Qz(t) H



.
.
− 2 z λ (t) − z(t), R`(t) − Qz λ (t) − R λ R`(t) − Qz λ (t) H
 .



.
.
.
.
.
.
2
2
2
≤ −2λ z λ (t) − z(t), z λ (t) H = λ kz(t)k H
− kz λ (t)k H
− kz λ (t) − z(t)k H
.
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By integrating this inequality and using the coercivity of Q, we obtain the desired inequality.
In order to prove the strong convergence of z λ to z in H 1 (0,T ; H ), we note that z λ → z in C([0,T ]; H )
.
.
and kz λ kL2 (0,T ;H) ≤ kzkL2 (0,T ;H) follow from the gained inequality. Hence, z λ * z in H 1 (0,T ; H ) and
the desired strong convergence follows from [8, Proposition 3.32].

Remark 3.6. The above proof shows that the inequality in (3.7) is by no means restricted to the specific
setting in (3.1), i.e., whenever ζ ∈ H 1 (0,T ; H ) and ζ λ ∈ H 1 (0,T ; H ) solve
.

.

ζ = A(Rд − Qζ ),

ζ (0) = ζ 0 ,

ζ λ = A λ (Rд − Qζ λ ), ζ λ (0) = ζ 0 ,
where A : H → 2 H is a maximal monotone operator, A λ : H → H its Yosida-approximation and
д ∈ L2 (0,T ; X) and ζ 0 ∈ H are given, then an inequality analogue to (3.7) holds (with ζ and ζ λ instead
of z and z λ ).
Since we are concerned with an optimal control problem with the external loads as control variable,
the continuity of the solution operator of (3.1) and its regularization w.r.t. variations in the external loads
is of particular interest, for instance when it comes to the existence of optimal controls, see Section 4
below. Since we aim to have a less restrictive control space in order to allow for as many control
functions as possible, the topology for the variations of the loads needed for our continuity results
should be as weak as possible. In particular, we aim to avoid strong convergence of (time-)derivatives
of the loads. The underlying idea is similar to [7, Theorem 3.16] and leads to the following
Lemma 3.7. Let {zn,0 }n ∈N ⊂ H and {`n }n ∈N ⊂ L2 (0,T ; X) be sequences such that zn,0 → z 0 in H and
`n → ` in L1 (0,T ; X). Assume further that {An }n ∈N is a sequence of maximal monotone operators such
that
An, λ (h) → Aλ (h)

(3.8)

for all λ > 0 and all h ∈ (R` − Qz λ )([0,T ]), as n → ∞, where z λ is the solution of (3.6) and An, λ denotes
the Yosida-approximation of An . Then, if a sequence {zn }n ∈N ⊂ H 1 (0,T ; H ) satisfies
.

zn (0) = zn,0 .

z n ∈ An (R`n − Qzn ),

(3.9)
.

and the derivatives z n are bounded in L2 (0,T ; H ), then zn * z in H 1 (0,T ; H ) and zn → z in C([0,T ]; H ).
Proof. Let λ > 0 be fixed, but arbitrary and define zn, λ ∈ H 1 (0,T ; H ) as solution of
.

z n, λ = An, λ (R`n − Qzn, λ ),

zn, λ (0) = zn,0 ,

whose existence and uniqueness can again be shown by Banach’s contraction principle as in case
of (3.6). Owing to [45, Proposition 55.2(b)], we obtain
.

.

kz λ (t) − z n, λ (t)k H ≤ kAλ (R`(t) − Qz λ (t)) − An, λ (R`(t) − Qz λ (t))k H
+ kAn, λ (R`(t) − Qz λ (t)) − An, λ (R`n (t) − Qzn, λ (t))k H
≤ kAλ (R`(t) − Qz λ (t)) − An, λ (R`(t) − Qz λ (t))k H
kQ kL(H;H)
kRkL(X;H)
+
kz λ (t) − zn, λ (t)k H +
k`(t) − `n (t)k X ,
λ
λ
and therefore, Gronwall’s inequality implies

kz λ − zn, λ kC([0,T ];H) ≤ C(λ) kz 0 − zn,0 k H + k` − `n kL1 (0,T ;X)

+ kAλ (R` − Qz λ ) − An, λ (R` − Qz λ )kL1 (0,T ;H) .
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Optimal control of an abstract evolution variational . . .

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The operators An, λ are uniformly Lipschitz continuous with Lipschitz constant λ−1 . Thus, thanks to
assumption (3.8), we can apply Lemma c.1 with M B (R` − Qz λ )[0,T ], N B H , G n B An, λ and
G B Aλ . Together with the assumptions on `n and zn,0 this gives that the right side in the inequality
above converges to zero as n → ∞. Using this, Proposition 3.5, and Remark 3.6 (with A = An ), we
conclude

(3.10)

lim sup kz − zn kC([0,T ];H) ≤ kz − z λ kC([0,T ];H) + lim sup kzn, λ − zn kC([0,T ];H)
n→∞
n→∞
s


λ .
.
kz kL2 (0,T ;H) + sup kz n kL2 (0,T ;H) .
≤
γQ
n ∈N
.

Now, since λ was arbitrary, (3.10) holds for every λ > 0. Therefore, as z n is bounded in L2 (0,T ; H )
by assumption, we obtain zn → z in C([0,T ]; H ). Moreover, again due to the boundedness assumption
.
on z n , there is a weakly converging subsequence in H 1 (0,T ; H ). Due to zn → z in C([0,T ]; H ), the
weak limit is unique and hence, the whole sequence zn converges weakly to z in H 1 (0,T ; H ).

Lemma 3.8. Let {λn }n ∈N ⊂ (0, ∞) be a sequence converging towards zero. Then the sequence An B Aλn ,
n ∈ N, of maximal monotone operators fulfills (3.8) for all λ > 0 and all h ∈ H .
Proof. At first we prove that, for all h ∈ H and 2λ > µ > 0, the following inequality holds
r
µ
(3.11)
kh − R λ (h)k H .
kR λ (h) − R λ+µ (h)k H ≤
2λ − µ
For this purpose, let h ∈ H be arbitrary and set y1 B R λ (h) and y2 B R λ+µ (h). Then we have
h−y
h−y
h ∈ y1 + λA(y1 ), hence, λ 1 ∈ A(y1 ) and analogously λ+µ2 ∈ A(y2 ). The monotonicity of A thus implies

µ

λ+µ
µ
2
2
0≤
+ kh − y1 k H
,
(h − y1 ) − (h − y2 ), y1 − y2
≤
− 1 ky1 − y2 k H
λ
2λ
2λ
H


hence,
2
ky1 − y2 k H
≤

µ
2
kh − y1 k H
,
2λ − µ

which yields (3.11). With this inequality and [45, Proposition 55.2 (d)] at hand, we obtain
(Aλn )λ (h) = Aλn +λ (h) =
which completes the proof.

1
1
(h − R λn +λ (h)) → (h − R λ (h)) = Aλ (h),
λn + λ
λ


Now, we are in the position to state our main convergence results in Theorem 3.10 and Theorem 3.11,
where the loads and initial data are no longer fixed. The first theorem addresses the continuity
properties of the solution operator to the original equation (3.1), whereas Theorem 3.11 deals with the
Yosida-approximation. In order to sharpen these convergence results and prove strong convergence in
H 1 (0,T ; H ), we additionally need the following
Assumption 3.9. The maximal monotone operator A is given as a subdifferential of a proper, convex
and lower semicontinuous function ϕ : H → (−∞, ∞], that is, A = ∂ϕ.
Theorem 3.10 (Continuity of the solution operator). Let {zn,0 }n ∈N ⊂ H and {`n }n ∈N ⊂ U(zn,0 , D(A))
be sequences such that zn,0 → z 0 in H , `n * ` in H 1 (0,T ; X) and `n → ` in L1 (0,T ; X). Moreover,
denote the solution of
.
z n ∈ A(R`n − Qzn ), zn (0) = zn,0
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Optimal control of an abstract evolution variational . . .

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by zn ∈ H 1 (0,T ; H ) (whose existence is guaranteed by Theorem 3.3). Then zn * z in H 1 (0,T ; H ) and
zn → z in C([0,T ]; H ).
If additionally `n → ` in H 1 (0,T ; X), A fulfills Assumption 3.9, and ϕ(R`n (0) − Qzn,0 ) → ϕ(R`(0) −
Qz 0 ), then zn → z in H 1 (0,T ; H ).
Proof. Thanks to Theorem 3.3, to be more precise (3.2), {zn } is bounded in C([0,T ]; H ). Since A0 is
.
bounded on bounded sets by assumption, (3.3) then gives that {z n } is bounded in L2 (0,T ; H ). Therefore,
we can apply Lemma 3.7 with An B A for all n ∈ N to obtain zn * z in H 1 (0,T ; H ) and zn → z in
C([0,T ]; H ).
If additionally `n → ` in H 1 (0,T ; X), A fulfills Assumption 3.9, and ϕ(R`n (0) − Qzn,0 ) → ϕ(R`(0) −
Qz 0 ), we can follow the lines of [24, Theorem 4.2 step 3)] to get
ˆ T
lim sup
n→∞

.

.

.

.

. .

.

(Q z n , z n ) H dt = lim sup −(R `n − Q z n , z n )L2 (0,T ;H) + (R `, z)L2 (0,T ;H)
n→∞

0

. .

= lim sup ϕ(R`n (0) − Qzn,0 ) − ϕ(R`n (T ) − Qzn (T )) + (R `, z)L2 (0,T ;H)
n→∞

. .

≤ ϕ(R`(0) − Qz 0 ) − ϕ(R`(T ) − Qz(T )) + (R `, z)L2 (0,T ;H)
ˆ T
. .
=
(Q z, z) H dt
0

where the secondpand last equation follows from [7, Lemme 3.3]. Hence, by equipping H with the
equivalent norm (Q·, ·) H , the strong convergence zn → z in H 1 (0,T ; H ) follows from [8, Proposition
3.32].

Theorem 3.11 (Convergence of the Yosida-approximation). The statement of Theorem 3.10 holds true
when zn is, for every n ∈ N, the solution of
.

(3.12)

z n = Aλn (R`n − Qzn ),

zn (0) = zn,0 ,

where {λn }n ∈N ⊂ (0, ∞) is a sequence converging to zero.
Proof. According to Lemma 3.8, the sequence of maximal monotone operators An B Aλn fulfills (3.8)
.
so that it only remains to prove that z n is bounded in L2 (0,T ; H ) to apply again Lemma 3.7. To this
end, let vn ∈ H 1 (0,T ; H ) be the solution of
.

v n ∈ A(R`n − Qvn ),

vn (0) = zn,0 ,

whose existence is guaranteed by Theorem 3.3 (note that `n ∈ U(zn,0 , D(A)) by assumption). Thanks
.
to Theorem 3.10, it holds vn * z in H 1 (0,T ; H ). From Proposition 3.5, it follows kz n kL2 (0,T ;H) ≤
.
.
kv n kL2 (0,T ;H) and consequently, z n is bounded in L2 (0,T ; H ). Thus, Lemma 3.7 yields zn * z in
H 1 (0,T ; H ) and zn → z in C([0,T ]; H ), as claimed.
If additionally `n → ` in H 1 (0,T ; X), A fulfills Assumption 3.9, and ϕ(R`n (0) − Qzn,0 ) → ϕ(R`(0) −
Qz 0 ), then Theorem 3.10 implies vn → z in H 1 (0,T ; H ) so that [8, Proposition 3.32] gives the strong
.
.
convergence zn → z in H 1 (0,T ; H ) because of kz n kL2 (0,T ;H) ≤ kv n kL2 (0,T ;H) as seen above.

Remark 3.12. The assertions of Theorem 3.10 and Theorem 3.11 are remarkable due to the following: As
a first approach to prove the (strong) convergence of the states in H 1 (0,T ; H ), one is tempted to follow
the lines of the proofs of [7, Lemme 3.1] and Proposition 3.5, respectively. This would however require
the strong convergence of the derivatives of the given loads, which we want to avoid in order to enable
less regular controls. The detour via the Yosida-regularization in Lemma 3.7 allows to overcome this
issue.
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Remark 3.13. If we would allow for more regular controls, then we could weaken the assumptions on
the maximal monotone operator A. For instance, if ` ∈ H 2 (0,T ; X), then we can drop the assumptions
that D(A) is closed and A0 is bounded on bounded sets. In this case, one can use [45, Theorem 55.A]
instead of [7, Proposition 3.4] in the proof of Theorem 3.3. The proof of [45, Theorem 55.A] also gives
.
the boundedness of z n in L∞ (0,T ; H ) in this case. Thus, Lemma 3.7 is again applicable and we can argue
similar as we did in the proof of Theorem 3.11 to verify the previous convergence results. In this setting,
we would not have any restrictions on A (except monotonicity), but would need more regular loads,
which is not favorable, as the latter serve as control variables in our optimization problem. Moreover, the
boundedness assumption on A is fulfilled for our concrete application problem in Section 7.4. Therefore,
we decided to choose the present setting and to impose the additional boundedness assumption on A.
Remark 3.14. It is to be noted that most of the above results can also be shown in more general BochnerSobolev spaces, that is, when loads are contained in W 1,r (0,T ; X) and states in W 1,r (0,T ; H ) for some
r ∈ [1, ∞). However, since a Hilbert space setting is advantageous when it comes to the derivation of
optimality conditions, we focus on the case r = 2.
Unfortunately, the Yosida-approximation is frequently not sufficient for the derivation of optimality
conditions by means of the standard adjoint approach, since the solution operator associated with (3.6)
is in general still not Gâteaux differentiable. Therefore, we apply a second regularization turning the
Yosida-approximation of A into a smooth operator. The properties needed to ensure convergence of
this second regularization are investigated in the following
Lemma 3.15 (Convergence of the Regularized Yosida-Approximation). Consider a sequence {λn }n ∈N ⊂
(0, ∞) and a sequence of Lipschitz continuous operators An : H → H , n ∈ N, such that
λn & 0 and

(3.13)

 T kQ k

1
L(H;H)
exp
sup kAn (h) − Aλn (h)k H → 0.
λn
λn
h ∈H

Let moreover {`n }n ∈N ⊂ C([0,T ]; X) be given and denote by zn , z λn ∈ C 1 ([0,T ]; H ) the solutions of
.

(3.14)
and

(3.15)

.

z n = An (R`n − Qzn ),

zn (0) = z 0 ,

z λn = Aλn (R`n − Qz λn ), z λn (0) = z 0 .

Then kzn − z λn kC 1 ([0,T ];H) → 0.
Proof. Again, thanks to the Lipschitz continuity of An and Aλn , the existence and uniqueness of zn
and zn, λ follows from Banach’s contraction principle by classical arguments. Moreover, the continuity
.
.
of `n carries over to the continuity of z n and z n, λ . Let us abbreviate c n B suph ∈H kAn (h) − Aλn (h)k H .
Then, in light of (3.14) and (3.15), we find
.

.

kz n (t) − z λn (t)k H ≤ c n +

kQ kL(H;H)
kzn (t) − z λn (t)k H
λn

∀ t ∈ [0,T ]

so that Gronwall’s inequality yields
.

.

kz n (t) − z λn (t)k H ≤

 kQ k


kQ kL(H;H) 
L(H;H)
T exp
T + 1 cn
λn
λn

∀ t ∈ [0,T ],


which completes the proof.

4 existence and approximation of optimal controls
Now we turn to the optimal control problem (P). We first address the existence of optimal solutions
and afterwards discuss the approximation of (P) in Section 4.2.
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4.1 existence of optimal controls
Based on Theorem 3.3, we reduce the optimal control problem (P) into a problem in the control variable
only. Recall that the control space Xc embeds compactly in X.
Definition 4.1. Let z 0 ∈ H and M ⊂ D(A). Due to Theorem 3.3, there exists for every ` ∈ H 1 (0,T ; Xc ) ∩
U(z 0 ; M) a solution z ∈ H 1 (0,T ; H ) of the state equation in (3.1). Consequently, we may define the
solution operator
S : H 1 (0,T ; Xc ) ∩ U(z 0 ; M) 3 ` → z ∈ H 1 (0,T ; H ).
This operator will be frequently called control-to-state map.
With the definition above, problem (P) is equivalent to the reduced problem:
(
(P)

⇐⇒

min

J (S(`), `),

s.t. ` ∈ H 1 (0,T ; Xc ) ∩ U(z 0 ; M).

Recall our standing assumptions on the objective, namely that J (z, `) = Ψ(z, `) + Φ(`), where
Ψ : H 1 (0,T ; W) × H 1 (0,T ; Xc ) → R is weakly lower semicontinuous and bounded from below and
Φ : H 1 (0,T ; Xc ) → R is weakly lower semicontinuous and coercive. These assumptions allow us to
show the existence of (globally) optimal solutions:
Theorem 4.2 (Existence of Optimal Solutions). Let z 0 ∈ H and M be a closed subset of D(A). Then there
exists a global solution of (P).
Proof. Based on Theorem 3.10, the proof follows the standard direct method of the calculus of variations. First of all, since Ψ is bounded from below and Φ is coercive, every infimal sequence of controls
is bounded in H 1 (0,T ; Xc ) and thus admits a weakly converging subsequence. Due to the compact
embedding of Xc in X, this sequence converges strongly in C([0,T ]; X) so that the weak limit belongs to U(z 0 ; M), due the closeness of M. Moreover, thanks to weak convergence in H 1 (0,T ; X) and
strong convergence in C([0,T ]; X), Theorem 3.10 gives weak convergence of the associated states in
H 1 (0,T ; H ). The weak lower semicontinuity of Ψ and Φ together with H ,→ W then implies the
optimality of the weak limit.

Clearly, in view of the nonlinear state equation, one cannot expect the optimal solution to be unique.
Note that, since D(A) is closed by our standing assumptions, the choice M = D(A) is feasible.

4.2 convergence of global minimizers
While the existence of optimal solutions for (P) can be shown by well-established techniques as seen
above, the derivation of optimality conditions is all but standard because of the lack of differentiability of
the control-to-state map. We therefore apply a regularization of A built upon the Yosida-approximation
in order to obtain a smooth control-to-state mapping. In view of Lemma 3.15, this regularization is
assumed to satisfy the following
Assumption 4.3. Let {An }n ∈N be a sequence of Lipschitz continuous operators from H to H such that,
together with a sequence {λn }n ∈N ⊂ (0, ∞), it holds
(4.1)

λn & 0 and

 T kQ k

1
L(H;H)
exp
sup kAn (h) − Aλn (h)k H → 0,
λn
λn
h ∈H

i.e., the requirements in Lemma 3.15 are fulfilled.
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In Section 7.4, we show how to construct such a regularization for a concrete application problem.
Given the regularization of A, we define the corresponding optimal control problem:






(Pn )

min J (z, `),
.

s.t. z = An (R` − Qz),






z(0) = z 0 ,



(z, `) ∈ H (0,T ; H ) × H 1 (0,T ; Xc ) ∩ U(z 0 ; M) .
1

Since An is Lipschitz continuous, the equation
.

z = An (R` − Qz),

z(0) = z 0

admits a unique solution for every z 0 ∈ H and every ` ∈ L2 (0,T ; X). Similar to Definition 4.1, we
denote the associated solution operator by
Sn : L2 (0,T ; X) → H 1 (0,T ; H ).
Moreover, the solution operator associated with the Yosida-approximation, i.e., the solution operator
.
of z = Aλn (R` − Qz), z(0) = z 0 , is denoted by
Sλn : L2 (0,T ; X) → H 1 (0,T ; H ).
Proposition 4.4 (Existence of Optimal Solutions of the Regularized Problems). Let n ∈ N, z 0 ∈ H , and
M a closed subset of D(A). Then, under Assumption 4.3, there exists a global solution of (Pn ).
Proof. Let `1 , `2 ∈ L2 (0,T ; X) be arbitrary and define zi B Sn (`i ), i = 1, 2. Then, due to the Lipschitz
continuity of An , we have for almost all t ∈ [0,T ]
.

.

kz 1 (t) − z 2 (t)k H = kAn (R`1 (t) − Qz 1 (t)) − An (R`2 (t) − Qz 2 (t))k H


≤ c kl 1 (t) − l 2 (t)k X + kz 1 (t) − z 2 (t)k H ,

(4.2)

which yields, thanks to Gronwall’s inequality , the Lipschitz continuity of Sn . Using this together with
the fact that Xc is compactly embedded into X, one can argue as in the proof of Theorem 4.2 to obtain
the existence of a global solution of (Pn ) for all n ∈ N.

Theorem 4.5 (Weak Approximation of Global Minimizers). Let z 0 ∈ H and M be a closed subset of D(A).
Suppose moreover that Assumption 4.3 holds and let {`n }n ∈N be a sequence of globally optimal controls
of (Pn ), n ∈ N. Then there exists a weak accumulation point and every weak accumulation point is a
global solution of (P).
Proof. Due to M ⊂ D(A), Proposition 3.5 gives Sλn (` 1 ) → S(` 1 ) in H 1 (0,T ; H ) so that Lemma 3.15
yields Sn (` 1 ) → S(` 1 ) in H 1 (0,T ; H ) and thus
lim sup Ψ(Sn (`n ), `n ) + Φ(`n ) = lim sup J (Sn (`n ), `n ) ≤ lim sup J (Sn (` 1 ), ` 1 ) = J (S(` 1 ), ` 1 ).
n→∞

n→∞

n→∞

Hence, by virtue of the boundedness of Ψ from below and the radial unboundedness of Φ, {`n } is
bounded and therefore admits a weak accumulation point in H 1 (0,T ; Xc ).
Let us now assume that a given subsequence of {`n }n ∈N , denoted by the same symbol for simplicity,
converges weakly to `˜ in H 1 (0,T ; Xc ). Since Xc is compactly embedded in X, we obtain `n → `˜ in
C([0,T ]; X) and consequently, `˜ ∈ U(z 0 ; M). In addition, the strong convergence in C([0,T ]; X) in
combination with Theorem 3.11 and Lemma 3.15 yields weak convergence of the states, i.e., Sn (`n ) *
˜ in H 1 (0,T ; H ) and thus also in H 1 (0,T ; W). Now, let ` be a global solution of (P). We can again
S(`)
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use Proposition 3.5 and Lemma 3.15 to obtain Sn (`) → S(`) in H 1 (0,T ; H ). This, together with the
weak lower semicontinuity of Ψ and Φ, implies
˜ `)
˜ = Ψ(S(`),
˜ `)
˜ + Φ(`)
˜ ≤ lim inf Ψ(Sn (`n ), `n ) + Φ(`n )
J (S(`),
n→∞

(4.3)

≤ lim sup J (Sn (`n ), `n ) ≤ lim sup J (Sn (`), `) = J (S(`), `),
n→∞

n→∞



giving in turn the optimality of the weak limit.

Corollary 4.6 (Strong Approximation of Global Minimizers). In addition to Assumption 4.3, assume
that Φ : H 1 (0,T ; Xc ) → R is such that, if a sequence {`n }n ∈N satisfies `n * ` in H 1 (0,T ; Xc ) and
Φ(`n ) → Φ(`), then `n → ` in H 1 (0,T ; Xc ). Then every weak accumulation point of a sequence of
globally optimal controls of (Pn ) is also a strong one.
Moreover, if in addition, at least one of the following holds
• Assumption 3.9 is satisfied, that is A = ∂ϕ, and ϕ is continuous on M or
• Ψ : H 1 (0,T ; W)×H 1 (0,T ; Xc ) → R is such that, if sequences {zn }n ∈N and {`n }n ∈N satisfy zn * z
in H 1 (0,T ; H ) and `n → ` in H 1 (0,T ; Xc ) and Ψ(zn , `n ) → Ψ(z, `), then zn → z in H 1 (0,T ; H ),
then the associated sequence of state also converges strongly in H 1 (0,T ; H ).
Proof. Consider an arbitrary accumulation point `˜ of a sequence of global minimizers of (Pn ), i.e.,
`n * `˜ in H 1 (0,T ; Xc ). From the previous proof, we know that then (4.3) holds, giving in turn
˜ `)
˜ + Φ(`).
˜
Ψ(Sn (`n ), `n ) + Φ(`n ) → Ψ(S(`),
˜ as seen in the previous proof, and both, Ψ and Φ, are weakly lower semicontinuous
Since Sn (`n ) * S(`),
˜ and Ψ(Sn (`n ), `n ) → Ψ(S(`),
˜ `).
˜ The hypothesis on Φ thus
by assumption, this implies Φ(`n ) → Φ(`)
1
˜
˜
yields `n → ` in H (0,T ; Xc ) so that ` is indeed a strong accumulation point as claimed.
Due to Xc ,→ X, the strong convergence carries over to H 1 (0,T ; X) and therefore, we deduce
˜ in H 1 (0,T ; H ), provided that Assumption 3.9 is fulfilled and
from Theorem 3.11 that Sλn (`n ) → S(`)
˜ − Qz 0 ) holds. If the additional requirements on Ψ are fulfilled, we also
ϕ(R`n (0) − Qz 0 ) → ϕ(R `(0)
˜ in H 1 (0,T ; H ), since we already showed `n → `˜ in
obtain the strong convergence Sλn (`n ) → S(`)
1
˜ in H 1 (0,T ; H ), which is the second
H (0,T ; Xc ). Thus, in both cases, Lemma 3.15 gives Sn (`n ) → S(`)
assertion.

Example 4.7. Let us assume that Xc is a Hilbert space. Then a possible objective functional fulfilling
the requirements on Φ in Corollary 4.6 reads as follows:
α
J (z, `) = Ψ(z, `) + k`kH2 1 (0,T ;Xc ) ,
2
2
1
i.e., Φ(`) B α/2 k`kH 1 (0,T ;X ) . Herein, Ψ : H (0,T ; H )×H 1 (0,T ; Xc ) → R is again lower semicontinuous
c
and bounded from below and α > 0 is a given constant. Since H 1 (0,T ; Xc ) is a Hilbert space, too, weak
convergence and norm convergence give strong convergence and consequently, this specific choice of
Φ fulfills the condition in Corollary 4.6.
Remark 4.8 (Approximation of Local Minimizers). By standard localization arguments, the above
convergence analysis can be adapted to approximate local minimizers. Following the lines of, for
instance, [12], one can show that, under the assumptions of Corollary 4.6, every strict local minimum
of (P) can be approximated by a sequence of local minima of (Pn ). A local minimizer ` of (P), which is
not necessarily strict, can be approximated by replacing the objective in (Pn ) by J (z, l) B J (z, l) + k` −
`kH 1 (0,T ;Xc ) , which is of course only of theoretical interest, cf. e.g. [4]. Since these results and their
proofs are standard, we omitted them.
Now that we answered the question of approximation of optimal controls via regularization, we
turn to the regularized problems and derive optimality conditions for these in the next two sections.
H. Meinlschmidt, C. Meyer, S. Walther

Optimal control of an abstract evolution variational . . .

J. Nonsmooth Anal. Optim. 1 (2020), 5800

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5 first-order optimality conditions
In the following, we consider a single element of the sequence of regularized problems. The associated
regularized operator is denoted by As so that the regularized optimal control problems reads as follows:

(Ps )











min J (z, `),
.

s.t. z = As (R` − Qz),

z(0) = z 0 ,



(z, `) ∈ H (0,T ; H ) × H 1 (0,T ; Xc ) ∩ U(z 0 ; M) .
1

Beside our standing assumption and the Lipschitz continuity required in Assumption 4.3, we need
the following additional assumptions for the derivation of first-order necessary optimality conditions
for (Ps ). Recall the continuous embeddings Y ,→ Z ,→ H from Section 2.
Assumption 5.1.
(i) J : H 1 (0,T ; W) × H 1 (0,T ; Xc ) → R is Fréchet differentiable.
(ii) As : Y → Y is Lipschitz continuous and Fréchet differentiable from Y to Z. Moreover, As0 (y)
can be extended to elements of L(Z; Z) and L(H ; H ), respectively, denoted by the same symbol. There exists a constant C such that these extensions satisfy kAs0 (y)zk Z ≤ C kz k Z and
kAs0 (y)hk H ≤ C khk H for all y ∈ Y, z ∈ Z, and h ∈ H .
Remark 5.2. It is well known that a norm gap is often indispensable to ensure Fréchet differentiability.
This is also the case in our application example in Section 7.4. This is the reason for considering two
different spaces Y and Z in context of the Fréchet differentiability of As in Assumption 5.1.
We start the derivation of optimality conditions for (Ps ) with the Fréchet-derivative of the associated
control-to-state mapping.

5.1 differentiability of the regularized control-to-state mapping
As As : Y → Y is supposed to be Lipschitz continuous and R and Q are not only linear and continuous
as operators with values in H , but also in Y according to our standing assumptions, Banach’s fixed
point theorem immediately implies that the state equation in (Ps ), i.e.,
.

(5.1)

z = As (R` − Qz),

z(0) = z 0 ,

admits a unique solution z ∈ H 1 (0,T ; Y) for every right hand side ` ∈ L2 (0,T ; X), provided that
z 0 ∈ Y. Therefore, similar to above, we can define the associated solution operator Ss : L2 (0,T ; X) →
H 1 (0,T ; Y) (for fixed z 0 ∈ Y). We will frequently consider this operator with different domains,
e.g. H 1 (0,T ; X), and ranges, in particular H 1 (0,T ; Z). With a little abuse of notation, these operators
are denoted by the same symbol.
Lemma 5.3 (Lipschitz Continuity of Ss ). The solution operator Ss is globally Lipschitz continuous from
L2 (0,T ; X) to H 1 (0,T ; Y).
Proof. This can be proven completely analogously to the Lipschitz continuity of Sn from L2 (0,T ; X)
to H 1 (0,T ; Y) in Proposition 4.4.

Lemma 5.4. Assume that Assumption 5.1(ii) is fulfilled and let y ∈ L2 (0,T ; Y) and w ∈ L2 (0,T ; Z) be
given. Then there exists a unique solution η ∈ H 1 (0,T ; Z) of
.

(5.2)
H. Meinlschmidt, C. Meyer, S. Walther

η = As0 (y)(w − Qη),

η(0) = 0.
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Proof. Let us define
B : [0,T ] × Z → Z,

(t, η) 7→ As0 (y(t))(w(t) − Qη)

.

so that (5.2) becomes η(t) = B(t, η(t)) a.e. in (0,T ), η(0) = 0. Now, given η ∈ L2 (0,T ; Z), [0,T ] 3
t 7→ B(t, η(t)) ∈ Z is Bochner measurable as a pointwise limit of Bochner measurable functions.
Furthermore, Assumption 5.1(ii) implies for almost all t ∈ [0,T ] and all η 1 , η 2 ∈ Z that kB(t, η 1 ) −
B(t, η 2 )k Z ≤ C kη 1 − η 2 k Z . Therefore, we can apply Banach’s fixed point argument to the integral
equation associated with (5.2), which gives the assertion.

Theorem 5.5 (Fréchet differentiability of the regularized solution operator). Under Assumption 5.1(ii),
the solution operator Ss is Fréchet differentiable from H 1 (0,T ; X) to H 1 (0,T ; Z). Its directional derivative
at ` ∈ H 1 (0,T ; X) in direction h ∈ H 1 (0,T ; X) is given by the unique solution of
.

η = As0 (R` − Qz)(Rh − Qη),

(5.3)

η(0) = 0,

where z B Ss (`) ∈ H 1 (0,T ; Y). Moreover, there exists a constant C such that kSs0 (`)hkH 1 (0,T ;Z) ≤
C khkL2 (0,T ;X) holds for all `, h ∈ H 1 (0,T ; X).
Proof. Let `, h ∈ H 1 (0,T ; X) be arbitrary and abbreviate zh B Ss (` + h). Thanks to Lemma 5.4, there
exists a unique solution η ∈ H 1 (0,T ; Z) of (5.3). Clearly, the solution operator of (5.3) is linear with
respect to h. Moreover, Assumption 5.1(ii) implies for almost all t ∈ [0,T ] that


.
kη(t)k Z ≤ C kh(t)k X + kη(t)k Z ,
so that Gronwall’s inequality gives kηkH 1 (0,T ;Z) ≤ C khkL2 (0,T ;X) , i.e., the continuity of the solution
operator of (5.3). This also proves the asserted inequality (after having proved that η = Ss0 (`)h, which
we do next).
It remains to verify the remainder term property. For this purpose, let us denote the remainder term
of As by r 1 , i.e.,
As (y + ζ ) = As (y) + As0 (y)ζ + r 1 (y; ζ )

with

kr 1 (y; ζ )k Z
→ 0 as ζ → 0 in Y.
kζ k Y

Moreover, we abbreviate
y B R` − Qz ∈ H 1 (0,T ; Y) and ζ B Rh − Q(zh − z) ∈ H 1 (0,T ; Y).
Then, in view of the definition of z, zh , and η (as solution of (5.3)), we find for almost all t ∈ [0,T ]
.

.

.

kz h (t) − z(t) − η(t)k Z = kAs (y(t) + ζ (t)) − As (y(t)) − As0 (y(t))(ζ (t) + Q(zh (t) − z(t) − η(t)))k Z
≤ kAs0 (y(t))Q(zh (t) − z(t) − η(t))k Z + kr 1 (y(t); ζ (t))k Z .
Hence, Assumption 5.1(ii) and Gronwall’s inequality yield
(5.4)

kzh − z − ηkH 1 (0,T ;Z) ≤ C kr 1 (y; ζ )kL2 (0,T ;Z) .

(note that r 1 (y; ζ ) ∈ L2 (0,T ; Z) by its definition as remainder term). Furthermore, thanks to Lemma 5.3
and the definition of ζ , we obtain
(5.5)
H. Meinlschmidt, C. Meyer, S. Walther

kζ kH 1 (0,T ;Y) ≤ C khkH 1 (0,T ;X)
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page 15 of 41

such that h → 0 in H 1 (0,T ; X) implies ζ → 0 in H 1 (0,T ; Y). The continuous embedding H 1 (0,T ; Y) ,→
C([0,T ]; Y) and the remainder term property of r 1 thus give for almost all t ∈ (0,T ) that
(5.6)

kr 1 (y(t); ζ (t))k Z
kr 1 (y(t); ζ (t))k Z kζ kH 1 (0,T ;Y)
≤C
→0
khkH 1 (0,T ;X)
kζ (t)k Y
khkH 1 (0,T ;X)

as h → 0 in H 1 (0,T ; X). Moreover, the Lipschitz continuity of As : Y → Y together with 5.1(ii),
Y ,→ Z, and (5.5) yield for almost all t ∈ (0,T ) that
k(As (y + ζ ) − As (y) − As0 (y)ζ )(t)k Z
kr 1 (y(t); ζ (t))k Z
kζ (t)k Y
=
≤C
≤ C.
khkH 1 (0,T ;X)
khkH 1 (0,T ;X)
khkH 1 (0,T ;X)
In combination with (5.6) and Lebesgue’s dominated convergence theorem, this yields
kr 1 (y; ζ )kL2 (0,T ;Z)
→0
khkH 1 (0,T ;X)
as h → 0 in H 1 (0,T ; X), which, in view of (5.4) finishes the proof.



Remark 5.6. It is to be noted that we did not employ the implicit function theorem to show the
.
differentiability of Ss . The reason is that H : z 7→ z − As (R` − Qz) is Fréchet differentiable from
H 1 (0,T ; Y) to L2 (0,T ; Z), but the derivative H 0(z) is not continuously invertible in these spaces, cf.
Lemma 5.4. On the other hand, H is not differentiable from H 1 (0,T ; Y) to L2 (0,T ; Y) (due to the
differentiability properties of As , see Remark 5.2), which would be the right spaces for the existence
result from Lemma 5.4. The same observation for a more abstract setting was already made in [41].

5.2 adjoint equation
Now that we know that the (regularized) control-to-state map is Gâteaux differentiable, we can apply
the standard adjoint approach to derive first-order necessary optimality conditions in form of a KarushKuhn-Tucker (KKT) system. To keep the discussion concise, we restrict our analysis to the case without
further control constraints. To be more precise, we require the following:
Assumption 5.7. Let z 0 ∈ Y such that −Qz 0 ∈ D(A). The set M in the definition of the set of admissible
controls is given by the singleton M = {−Qz 0 } such that

 
U B U z 0 ; {−Qz 0 } = ` ∈ H 1 (0,T ; X) : `(0) ∈ ker R .
Note that U is a linear subspace of H 1 (0,T ; X).
Remark 5.8 (Additional Control Constraints). One could allow for additional control constraints in
our analysis, even more complex ones than the ones covered by U(z 0 ; M) such as for instance box
constraints over the whole time interval or vanishing initial and final loading, i.e., `(0) = `(T ) = 0, which
is certainly meaningful for many practically relevant problems. However, since the differentiability of
the control-to-state map is the essential issue in the derivation of optimality conditions and additional
(convex and closed) control constraints can be incorporated by standard argument, we decided to leave
them out in order to keep the discussion as concise as possible.
However, without any further assumptions, the existence of solutions to the unregularized state
equation (3.1) cannot be guaranteed. To be more precise, one needs that R`(0) − Qz 0 ∈ D(A), see
Theorem 3.3, which holds in case of U, provided that −Qz 0 ∈ D(A). This is the reason for considering
the set H 1 (0,T ; XC ) ∩ U as set of admissible controls in the rest of the paper.
Note moreover that, if the operator R is injective (which is the case in Section 7), then U = {` ∈
1
H (0,T ; X) : `(0) = 0}.
H. Meinlschmidt, C. Meyer, S. Walther

Optimal control of an abstract evolution variational . . .

J. Nonsmooth Anal. Optim. 1 (2020), 5800

page 16 of 41

The chain rule immediately gives that the reduced objective defined by
F : H 1 (0,T ; Xc ) → R,

(5.7)

` 7→ J (Ss (`), `)

is Fréchet differentiable, too. Thus, by standard arguments, one derives the following
Lemma 5.9 (Purely Primal Necessary Optimality Conditions). Let Assumption 5.1 and Assumption 5.7
hold. Then, if a control ` ∈ H 1 (0,T ; Xc ) ∩ U with associated state z = Ss (`) is locally optimal for (Ps ),
then
F 0(`)h = Jz0 (z, `)Ss0 (`)h + J`0 (z, `)h = 0,

(5.8)
for all h ∈ H 1 (0,T ; Xc ) ∩ U.

Next, we reformulate (5.8) in terms of a KKT-system by introducing an adjoint equation, which
formally reads
.

φ = QAs0 (y)∗φ + v,

(5.9)

φ(T ) = 0.

Depending on the regularity of the right hand side v, we define different notions of solutions:
Definition 5.10. Let y ∈ L2 (0,T ; Y) and v ∈ H 1 (0,T ; H )∗ be given. A function φ ∈ L2 (0,T ; H ) is called
weak solution of (5.9), if


.

(5.10)
− φ, η L2 (0,T ;H) = φ, As0 (y)Qη L2 (0,T ;H) + v(η)
holds for all η ∈ H 1 (0,T ; H ) with η(0) = 0.
If v takes the form
v(η) = (v 1 , η)L2 (0,T ;H) + (v 2 , η(T )) H

(5.11)

with some v 1 ∈ L2 (0,T ; H ) and v 2 ∈ H , then we call φ ∈ H 1 (0,T ; H ) strong solution of (5.9), if, for
almost all t ∈ (0,T ),


.
(5.12)
φ(t) = QAs0 (y)∗φ (t) + v 1 (t) in H, φ(T ) = −v 2 in H .
In the following, we will—as usual—identify v ∈ L2 (0,T ; H ) with an element of H 1 (0,T ; H )∗ via
(v, · )L2 (0,T ;H) and denote this element with a slight abuse of notation by the same symbol.
Lemma 5.11. Let y ∈ L2 (0,T ; Y) and v ∈ H 1 (0,T ; H )∗ . Then there is a unique weak solution of (5.9),
which is given by φ B −v ◦ Sy ∈ L2 (0,T ; H )∗ = L2 (0,T ; H ), where Sy : L2 (0,T ; H ) → H 1 (0,T ; H ) is
the solution operator of
.

η = −As0 (y)Qη + w,

(5.13)

η(0) = 0,

that is, Sy (w) = η.
Moreover, if v is of the form (5.11), then there exists a unique strong solution of (5.9), and the weak and
the strong solution coincide.
Proof. At first note that the existence of a solution of (5.13) can be proven exactly as in Lemma 5.4.
.
Let η ∈ H 1 (0,T ; H ) with η(0) = 0 be arbitrary and define w B η + As0 (y)Qη ∈ L2 (0,T ; H ), hence,
η = Sy (w). By the definition of w and φ, it follows that
.

(η + As0 (y)Qη, φ)L2 (0,T ;H) = (φ, w)L2 (0,T ;H) = −v(Sy (w)) = −v(η),
i.e., (5.10) holds. Since η was arbitrary, we see that φ is a weak solution of (5.9).
H. Meinlschmidt, C. Meyer, S. Walther

Optimal control of an abstract evolution variational . . .

J. Nonsmooth Anal. Optim. 1 (2020), 5800

page 17 of 41

To prove uniqueness, let φ̃ ∈ L2 (0,T ; H ) be another weak solution. Then, we choose an arbitrary
w ∈ L2 (0,T ; H ) and set η B Sy (w) to see that


.
(φ, w)L2 (0,T ;H) = −v(η) = φ̃, η + As0 (y)Qη L2 (0,T ;H) = (φ̃, w)L2 (0,T ;H) ,
and therefore φ = φ̃.
Now we turn to the strong solution and suppose that v is as given in (5.11). Existence and uniqueness
of a strong solution can again be shown by means of Banach’s fixed point theorem. To this end, let us
consider the affine-linear operator
B : [0,T ] × Z → Z,

B(t, φ) = QAs0 (y(t))∗φ + v 1 (t).

Since H is separable by our standing assumptions, we can apply [20, Chap. IV, Thm. 1.4] to obtain that,
for every φ ∈ L2 (0,T ; H ), the mapping (0,T ) 7→ B(t, φ(t)) is Bochner measureable. Moreover, since
kAs0 (y)∗ kL(H;H) = kAs0 (y)kL(H;H) , Assumption 5.1(ii) yields that B is also Lipschitz continuous w.r.t. the
second variable for almost all t ∈ (0,T ). Therefore, similarly to the proof of Lemma 5.4, one can apply
Banach’s fixed point theorem to the integral equation associated with (5.12) to establish the existence
of a unique strong solution.
Finally, if we test (5.12) with an arbitrary η ∈ H 1 (0,T ; H ) with η(0) = 0 and integrate by parts, then
we see that every strong solution is also a weak solution. Since the latter one is unique, as seen above,
we deduce that weak and strong solution coincide.

Theorem 5.12 (KKT-Conditions for (Ps )). Assume that Assumption 5.1 and Assumption 5.7 hold and let
` ∈ H 1 (0,T ; Xc ) ∩ U be a locally optimal control for (Ps ) with associated state z = Ss (`). Then there
exists a unique adjoint state φ ∈ L2 (0,T ; H ) such that the following optimality system is fulfilled
.

(5.14a)
(5.14b)

(5.14c)

z = As (R` − Qz),
.



 − φ, η L2 (0,T ;H)



0

+ Jz0 (z, `)η
 = φ, As (R` − Qz)Qη 2
L
(0,T
;H)



φ, As0 (R` − Qz)Rh 2
= J`0 (z, `)h
L (0,T ;H)

z(0) = z 0 ,
∀ η ∈ H 1 (0,T ; H ) :
η(0) = 0
∀ h ∈ H 1 (0,T ; Xc ) ∩ U.

If J enjoys extra regularity, namely
(5.15)

J (z, `) = Ψ1 (z, `) + Ψ2 (z(T ), `(T )) + Φ(`)

with two Fréchet differentiable functionals Ψ1 : L2 (0,T ; H ) × H 1 (0,T ; Xc ) → R and Ψ2 : H × Xc → R,
then φ ∈ H 1 (0,T ; H ) is a strong solution of
(5.16)



∂Ψ1
.
φ(t) = QAs0 (R` − Qz)∗φ (t) +
(z, `),
∂z

φ(T ) = −

∂Ψ2
(z(T ), `(T )).
∂z

Remark 5.13. The exemplary objective functionals in Section 7 are precisely of the form in (5.15).
Proof of Theorem 5.12. Since Jz0 (z, `) ∈ H 1 (0,T ; W)∗ ,→ H 1 (0,T ; H )∗ , Lemma 5.11 gives the existence
of a unique solution of (5.14b). Now, let h ∈ H 1 (0,T ; Xc ) ∩ U be arbitrary and define η B Ss0 (`)h ∈
H 1 (0,T ; Z) ⊂ H 1 (0,T ; H ). The weak form of the adjoint equation then implies


 .

= φ, η + As0 (R` − Qz)Qη 2
= −Jz0 (Ss (`), `)η.
φ, As0 (R` − Qz)Rh 2
(5.17)
L (0,T ;H)

L (0,T ;H)

This together with Lemma 5.9 shows that (z, `, φ) fulfills the optimality system (5.14). If J is of the form
in (5.15), then Lemma 5.11 implies that the weak solution of the adjoint equation is in fact a strong
solution and solves (5.16).

H. Meinlschmidt, C. Meyer, S. Walther

Optimal control of an abstract evolution variational . . .

J. Nonsmooth Anal. Optim. 1 (2020), 5800

page 18 of 41

Corollary 5.14. Let Assumption 5.1 and Assumption 5.7 hold. Then `¯ ∈ H 1 (0,T ; Xc ) ∩ U with associated
state z = Ss (`) fulfills (5.8) if and only if there exists an adjoint state φ ∈ L2 (0,T ; H ) such that (z, `, φ)
satisfies the optimality system (5.14).
Proof. The proof of Theorem 5.12 already shows that (5.8) implies the optimality system in (5.14).
To prove the reverse implication, assume that (z, `, φ) fulfills the optimality system (5.14). Then
choose an arbitrary h ∈ H 1 (0,T ; H ), define η B Ss0 (`)h, and use the fact that φ is the weak solution
of (5.14b) to obtain (5.17). This together with (5.14c) finally give (5.8).

Example 5.15. Under suitable additional assumptions, it is possible to further simplify the gradient
equation (5.14c). For this purpose assume that R is injective (so that U = {` ∈ H 1 (0,T , X) : `(0) = 0}),
Xc is a Hilbert space, and
J (z, `) = Ψ1 (z, `) + Ψ2 (z(T ), `(T )) +

(5.18)

γ . 2
k `kL2 (0,T ;Xc ) ,
2

where Ψ1 : H 1 (0,T ; W) × L2 (0,T ; Xc ) → R and Ψ2 : W × Xc → R are Fréchet differentiable and γ > 0.
This type of objective will also appear in the application problem in Section 7. Then (5.14c) becomes

(5.19)

γ (∂t `, ∂t h)L2 (0,T ;Xc ) −

ˆ T

hR ∗As0 (R` − Qz)∗φ, hi X ∗, X dt

ˆ T
0

+

0

∂Ψ1
∂Ψ2
(z, `)h dt +
(z(T ), `(T ))h(T ) = 0
∂`
∂`
∀ h ∈ H 1 (0,T ; Xc ) with h(0) = 0,

where we identified ∂` Ψ1 (z, `) ∈ (L2 (0,T ; Xc ))∗ = L2 (0,T ; Xc ). Note that Xc as a Hilbert space satisfies
the Radon-Nikodým-property. Since Xc ,→ X, we may identify R ∗As0 (R` − Qz)∗φ with an element
of L2 (0,T ; Xc ), too, which we denote by the same symbol. Then, if we choose h(t) = ψ (t) ξ with
ψ ∈ Cc∞ (0,T ) and ξ ∈ Xc arbitrary, we obtain


−

ˆ T h
i

∂Ψ1
γ ∂t ψ ∂t ` + R ∗As0 (R` − Qz)∗φ ψ −
(z, `)ψ dt, ξ
= 0.
Xc
∂`
0

Now, since ξ ∈ Xc was arbitrary, we find that the second distributional time derivative of ` is a regular
distribution in L2 (0,T ; Xc ), i.e., ` ∈ H 2 (0,T ; Xc ), satisfying for almost all t ∈ (0,T )
(5.20)


∗
∂Ψ1
γ ∂t2 `(t) + R ∗As0 R`(t) − Qz(t) φ(t) =
(z, `)(t)
∂`

in Xc .

Since Xc is supposed to be a Hilbert space, we can apply integration by parts to (5.19). Together with
` ∈ U = {` ∈ H 1 (0,T ; X) : `(0) = 0} and (5.20), this implies the following boundary conditions:
(5.21)

`(0) = 0,

γ ∂t `(T ) = −

∂Ψ2
(z(T ), `(T )),
∂`

where we again identified ∂` Ψ2 (z(T ), `(T )) ∈ Xc∗ with its Riesz representative. In summary, we have
thus seen that the gradient equation in (5.14c) becomes an operator boundary value problem in Xc ,
namely (5.20)–(5.21).
H. Meinlschmidt, C. Meyer, S. Walther

Optimal control of an abstract evolution variational . . .

J. Nonsmooth Anal. Optim. 1 (2020), 5800

page 19 of 41

6 second-order sufficient conditions
The next section is devoted to the derivation of second-order sufficient optimality conditions for the
regularized problem (Ps ). As it was the case for the first-order conditions, the main part concerns
the differentiability properties of the control-to-state map Ss and the reduced objective, to be more
precise to show that these are twice continuously Fréchet differentiable. For this purpose, we need the
following sharpened assumptions on the objective and the regularized operator As :
Assumption 6.1.
(i) J : H 1 (0,T ; W) × H 1 (0,T ; Xc ) → R is twice continuously Fréchet differentiable.
(ii) The Fréchet-derivative As0 is Lipschitz continuous from Y to L(Z; Z). Moreover, for every
y ∈ Y, As0 (y) can be extended to an element of L(W; W). The mapping arising in this way is
Lipschitz continuous from Y to L(W; W). Furthermore, there is a constant C > 0 such that
kAs0 (y)w k W ≤ C kw k W hold for all y ∈ Y and all w ∈ W.
(iii) As0 is Fréchet differentiable from Y to L(Z; W). For all y ∈ Y, its derivative As00(y) can be
extended to an element of L(Z; L(Z; W)) and the mapping y 7→ As00(y) is continuous in these
spaces. Moreover, there exists a constant C such that kAs00(y)[z 1 , z 2 ]k W ≤ C kz 1 k Z kz 2 k Z for all
y ∈ Y and all z 1 , z 2 ∈ Z.
Remark 6.2. We point out that a second norm gap arises in Assumption 6.1, since As0 is only Fréchet
differentiable as an operator with values in W and not in Z ,→ W. This assumption is again motivated
by the application problem in Section 7. The example given there demonstrates that such as second
norm gap is indeed necessary in general, since, given a concrete application, one cannot expect As to
be twice Fréchet differentiable in Y, and even not as an operator from Y to Z.
The following proposition addresses the second derivative of the solution operator under the above
assumptions. Its proof is in principle completely along the lines of the proof of Theorem 5.5 on the
first derivative of S. We therefore postpone it to Appendix a.
Proposition 6.3 (Second Derivative of the Solution Operator). Under Assumption 5.1(ii) and Assumption 6.1(ii) & (iii), the solution operator Ss : H 1 (0,T ; X) → H 1 (0,T ; W) is twice Fréchet differentiable.
Given `, h 1 , h 2 ∈ H 1 (0,T ; X), its second derivative Ss00(`)[h 1 , h 2 ] ∈ H 1 (0,T ; W) is given by the unique
solution of
.

(6.1)

ξ = As00(R` − Qz)[Rh 1 − Qη 1 , Rh 2 − Qη 2 ] − As0 (R` − Qz)Qξ ,

ξ (0) = 0,

where z B Ss (`) ∈ H 1 (0,T ; Y) and ηi B Ss0 (`)hi ∈ H 1 (0,T ; Z), i = 1, 2.
Moreover, there exists a constant C such that
(6.2)

kSs00(`)[h 1 , h 2 ]kH 1 (0,T ;W) ≤ C kh 1 kH 1 (0,T ;X) kh 2 kH 1 (0,T ;X)

for all `, h 1 , h 2 ∈ H 1 (0,T ; X).
Lemma 6.4. Assume that Assumption 5.1 (ii) and Assumption 6.1 (ii) and (iii) are fulfilled. Then there
exists a constant C such that
kSs00(`1 ) − Ss00(`2 )kL(H 1 (0,T ;X);L(H 1 (0,T ;X);H 1 (0,T ;W)))
≤ C kAs00(R`1 − Qz 1 ) − As00(R`2 − Qz 2 )kL2 (0,T ;L(Z;L(Z;W))) + k`1 − `2 kH 1 (0,T ;X)




holds for all `1 , `2 ∈ H 1 (0,T ; X), where zi B Ss (`i ), i = 1, 2.
H. Meinlschmidt, C. Meyer, S. Walther

Optimal control of an abstract evolution variational . . .

J. Nonsmooth Anal. Optim. 1 (2020), 5800

page 20 of 41

Proof. Let `1 , `2 , h 1 , h 2 ∈ H 1 (0,T ; X) be arbitrary. We again abbreviate zi B Ss (`i ), ηi, j B Ss0 (`i )h j ,
ξ i B Ss00(`i )[h 1 , h 2 ], and yi B R`i − Qzi for i, j ∈ {1, 2}. By the equation for Ss00, we obtain for almost
all t ∈ [0,T ]
.

.

ξ 1 − ξ 2 = As00(y1 )[Rh 1 − Qη 1,1 , Rh 2 − Qη 1,2 ] − As0 (y1 )Qξ 1
− As00(y2 )[Rh 1 − Qη 2,1 , Rh 2 − Qη 2,2 ] − As0 (y2 )Qξ 2


= As00(y1 )(Rh 1 − Qη 1,1 ) − As00(y2 )(Rh 1 − Qη 2,1 ) (Rh 2 − Qη 1,2 )


+ As00(y2 )[Rh 1 − Qη 2,1 , Q(η 2,2 − η 1,2 )] + As0 (y2 ) − As0 (y1 ) Qξ 1 + As0 (y2 )Q(ξ 2 − ξ 1 ).
With the help of
As00(y1 )(Rh 1 − Qη 1,1 ) − As00(y2 )(Rh 1 − Qη 2,1 )


= As00(y1 ) − As00(y2 ) (Rh 1 − Qη 1,1 ) + As00(y2 )Q(η 2,1 − η 1,1 ),
and Gronwall’s inequality, we thus arrive at
kξ 1 − ξ 2 kH 1 (0,T ;W)
h
≤ C kRh 1 − Qη 2,1 kH 1 (0,T ;Z) kη 1,2 − η 2,2 kH 1 (0,T ;Z) + ky1 − y2 kH 1 (0,T ;Y) kξ 1 kH 1 (0,T ;W)

+ kAs00(y1 ) − As00(y2 )kL2 (0,T ;L(Z;L(Z;W))) kRh 1 − Qη 1,1 kH 1 (0,T ;Z)

i
+ kη 1,1 − η 2,1 kH 1 (0,T ;Z) kRh 2 − Qη 1,2 kH 1 (0,T ;Z)


≤ C kAs00(y1 ) − As00(y2 )kL2 (0,T ;L(Z;L(Z;W))) + k`1 − `2 kH 1 (0,T ;X) kh 1 kH 1 (0,T ;X) kh 2 kH 1 (0,T ;X) ,
where we used the estimate in Theorem 5.5, (6.2), and the Lipschitz continuity of Ss0 by Lemma a.1 in
the Appendix.

If As00 were Lipschitz continuous from Y to L(Z; L(Z; W)), then Lemma 6.4 would immediately
imply the Lipschitz continuity of Ss00. However, to obtain the continuity of the second derivative, this
additional assumption is not necessary as the following theorem shows:
Theorem 6.5 (Second-Order Continuous Fréchet Differentiability of the Solution Operator). Suppose
that Assumption 5.1(ii) and Assumption 6.1(ii) & (iii) are fulfilled. Then Ss : H 1 (0,T ; X) → H 1 (0,T ; W)
is twice continuously Fréchet differentiable. Its second derivative at ` ∈ H 1 (0,T ; X) in directions h 1 , h 2 ∈
H 1 (0,T ; X) is given by the unique solution of (6.1).
Proof. Thanks to Proposition 6.3, we only have to show that Ss00 is continuous from H 1 (0,T ; X) to
L(H 1 (0,T ; X); L(H 1 (0,T ; X); H 1 (0,T ; W))). For this let {`n }n ∈N ⊂ H 1 (0,T ; X) and ` ∈ H 1 (0,T ; X) be
given such that `n → ` in H 1 (0,T ; X) so that in particular `n → ` in C([0,T ]; X). Then, Lemma 5.3
implies zn B Ss (`n ) → Ss (`) C z in C([0,T ]; Y). With this convergence results at hand, we can apply
Lemma c.2 with M = [0,T ], N = Y, G n = R`n − Qzn and G = R` − Qz to see that
U B

∞
Ø

 

(R`n − Qzn )([0,T ]) ∪ (R` − Qz)([0,T ])

n=1

is compact. Therefore, thanks to the continuity assumption in Assumption 6.1(iii),
As00 : Y → L(Z; L(Z; W)) is uniformly continuous on U . Consequently, As00(R`n − Qzn ) converges to
As00(R` − Qz) in C([0,T ]; L(Z; L(Z; W))), which, together with Lemma 6.4, yields the assertion. 
H. Meinlschmidt, C. Meyer, S. Walther

Optimal control of an abstract evolution variational . . .

J. Nonsmooth Anal. Optim. 1 (2020), 5800

page 21 of 41

Remark 6.6. It is to be noted that the regularized state equation (5.1) and the equations corresponding
to the derivatives of Ss , i.e., (5.3) and (6.1), provide more regular solutions under the hypotheses of
Assumption 5.1(ii) and Assumption 6.1(ii) & (iii). Indeed, if `, h 1 , h 2 ∈ H 1 (0,T ; X), then the solutions of
all three equations can be shown to be continuously differentiable in time with values in the respective
spaces (Y, Z, and W, respectively). Moreover, the time derivatives of z and η are absolutely continuous
and the same would hold for ξ , if As00 were Lipschitz continuous. However, we did not exploit this
additional regularity, since the original unregularized problem (P) does not provide this property in
general.
With the above differentiability result at hand, it is now standard to derive the following:
Theorem 6.7 (Second-Order Sufficient Optimality Conditions for (Ps )). Assume that Assumption 5.1,
Assumption 5.7, and Assumption 6.1 hold. Let (z, `, φ) ∈ H 1 (0,T ; Y) × (H 1 (0,T ; Xc ) ∩ U) × L2 (0,T ; H )
be a solution of the optimality system (5.14). Moreover, suppose that there is a δ > 0 such that
F 00(`)h 2 ≥ δ khkH2 1 (0,T ;Xc )

(SSC)

for all h ∈ H 1 (0,T ; Xc ) ∩ U, where F is the reduced objective from (5.7). Then (z, `) is locally optimal for
(Ps ) and there exist ε > 0 and τ > 0 such that the following quadratic growth condition
F (`) ≥ F (`) + τ k` − `kH2 1 (0,T ;Xc )

(6.3)

holds for all ` ∈ H 1 (0,T ; Xc ) ∩ U with k` − `kH 1 (0,T ;Xc ) ≤ ε.
Proof. Thanks to the assumptions on J and Theorem 6.5, the chain rule implies that the reduced
objective function F (·) = J (Ss (·), ·) : H 1 (0,T ; Xc ) → R is twice continuously Fréchet differentiable and,
according to Corollary 5.14, the equation in (5.8) holds for all h ∈ H 1 (0,T ; Xc ) ∩ U. Since U is a linear
subspace, the claim then follows from standard arguments, see e.g. [39, Satz 4.23].

Remark 6.8. As already mentioned in Remark 5.8, one could also account for additional control constraints. In this case, a critical cone would arise in the second-order conditions, cf. e.g. the survey
article [13].
Using the adjoint equation, the second derivative of the reduced objective in (SSC) can be reformulated as follows:
Corollary 6.9. Assume in addition to the hypotheses of Assumption 6.1(iii) that kAs00(y)[z 1 , z 2 ]k H ≤
C kz 1 k Z kz 2 k Z for all y ∈ Y and z 1 , z 2 ∈ Z, i.e., the last inequality in Assumption 6.1 holds in H
instead of the weaker space W. Then it holds for all `, h ∈ H 1 (0,T ; H ) that


F 00(`)h 2 = Ψ 00(z, `)(η, h)2 + Φ00(`)h 2 − φ, As00(R` − Qz)(Rh − Qη)2 L2 (0,T ;H) ,
where z = Ss (`), η = Ss0 (`)h, and φ solves the adjoint equation in (5.14b).
Proof. Let us again abbreviate y = R` − Qz. According to the chain rule, the second derivative of the
reduced objective is given by
2
2
∂
∂
∂
∂
F 00(`)h 2 = ∂`
2 J (z, `)h + ∂z 2 J (z, `)η + 2 ∂`∂z J (z, `)[h, η] + ∂z J (z, `)ξ
2

2

2

∂
= Ψ 00(z, `)(η, h)2 + Φ00(`)h 2 + ∂z
J (z, `)ξ

with z = Ss (`), η = Ss0 (`)h, and ξ = Ss00(`)h 2 . Now, since As00(y) is a bilinear form on H by assumption,
we obtain that ξ ∈ H 1 (0,T ; H ). Therefore, we are allowed to test the adjoint equation in (5.14b) (in its
weak form) with ξ , which results in
.
∂
J
(z,
`)ξ
=
−(φ,
ξ
+ As0 (y)Qξ )L2 (0,T ;H) = −(φ, As00(y)(Rh − Qη)2 )L2 (0,T ;H) ,
∂z

where we used the precise form of Ss00(`) in (6.1) for the last identity.
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7 application to optimal control of homogenized elastoplasticity
In the upcoming sections, we apply the analysis from the previous sections to an optimal control
problem governed by a system of equations that arise as homogenization limit in elastoplasticity and
was derived in [33, Theorem. 2.2]. It describes the evolution of plastic deformation in a material with
periodic microstructure and formally (i.e., in its strong form) reads as follows:
(7.1a)

−∇x · π Σ = f

(7.1b)
(7.1c)

in Ω,

Σ = C(∇xs u + ∇sy v − Bz)
−∇y · Σ = 0
.

in Ω × Y ,
in Ω × Y ,

(7.1d)

z ∈ A(B Σ − Bz)

in Ω × Y ,

(7.1e)

u=0

on ΓD ,

(7.1f)

ν · πΣ = д

on ΓN ,

(7.1g)

z(0) = z 0

in Ω × Y .

>

Herein, Ω ⊂ Rd , d = 2, 3, is a given domain occupied by the body under consideration, while Y = [0, 1]d
is the unit cell. The boundary of Ω consists of two disjoint parts, the Dirichlet boundary ΓD and the
Neumann boundary ΓN . Furthermore, u : (0,T ) × Ω → Rd is the displacement on the macro level, while
v : (0,T ) × Ω × Y → Rd is the displacement reflecting the micro structure. The stress tensor is denoted
×d and z : (0,T ) × Ω × Y → V is the internal variable describing changes in
by Σ : (0,T ) × Ω × Y → Rdsym
the material behavior under plastic deformation (such as hardening), where V is a finite dimensional
Banach space. Moreover, ∇xs B 21 (∇x + ∇x> ) is the linearized strain in Ω and ∇sy is defined analogously.
×d ) and the hardening parameter B : Ω × Y → L(V) are given
The elasticity tensor C : Ω × Y → L(Rdsym
×d ), one recovers the plastic strain from
linear and coercive mappings and, by B : Ω × Y → L(V; Rdsym
the internal variables z. The evolution of the internal variables is determined by a maximal monotone
operator A: V → 2V . In Section 7.4 below, we present a concrete example for such an operator, namely
the case of linear kinematic hardening with von Mises yield condition. Finally, z 0 is a given initial state
and π is the averaging over the unit cell, i.e.,
ˆ
1
Σ(·, y) dy .
(7.2)
π : Σ 7→
Σ(·, y) dy B
|Y | Y
Y
The precise assumptions on these data as well as the precise notion of solutions to (7.1) are given below.
The volume force f : (0,T ) × Ω → Rd and the boundary loads д : (0,T ) × ΓN → Rd , serve as control
variables. In the following, we will frequently write ` for the tuple (f , д). Possible objectives could
include a desired displacement or stress distribution at end time, i.e.,
ˆ
ˆ
β
α
|u(T ) − ud | 2 dx +
|(π Σ)(T ) − σd | 2 dx + Φ(`),
J (u, Σ, `) B
2 Ω
2 Ω
×d are given desired displacement and stress field, respectively,
where ud : Ω → Rd and σd : Ω → Rdsym
α, β ≥ 0, and Φ is a regularization term depending on the choice of the control space that will be
specified below, see Remark 7.14.

7.1 homogenized plasticity – notation and standing assumptions
Before discussing the optimal control problem, we first have to introduce the precise notion of solution
for homogenized elastoplasticity system in (7.1). For this purpose, we need several assumptions and
definitions. We start with the following
Assumption 7.1 (Hypotheses on the data in (7.1)).
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• Regularity of the domain: The domain Ω ⊂ Rd , d ∈ {2, 3}, is bounded with Lipschitz boundary
Γ. The boundary consists of two disjoint measurable parts ΓN and ΓD such that Γ = ΓN ∪ ΓD .
While ΓN is a relatively open subset, ΓD is a relatively closed subset of Γ with positive measure.
In addition, the set Ω ∪ ΓN is regular in the sense of Gröger, cf. [23].
• Assumptions on the coefficients: The elasticity tensor and the hardening parameter satisfy C ∈
×d )) and B ∈ L∞ (Ω × Y ; L(V)) and are symmetric and uniformly coercive, i.e.,
L∞ (Ω × Y ; L(Rdsym
there exist constants c > 0 and b > 0 such that
C(x, y)σ : σ ≥ c kσ kR2 d ×d

×d
∀ σ ∈ Rdsym
, f.a.a. (x, y) ∈ Ω × Y ,

B(x, y)ζ : ζ ≥ b kζ kV2

∀ ζ ∈ V,

f.a.a. (x, y) ∈ Ω × Y .

×d )) is a given linear mapping.
In addition, B ∈ L∞ (Ω × Y ; L(V; Rdsym

Next, we define the function spaces for the various variables in (7.1):
Definition 7.2 (Function spaces). Let s ∈ [1, ∞). For the quantities in (7.1), we define the following
spaces:
• space for the macro displacement u:
W 1, s (Ω;Rd )

U s B WD1,s (Ω; Rd ) B ψ |Ω : ψ ∈ C 0∞ (Rd ; Rd ), supp(ψ ) ∩ ΓD = ∅

• space for the internal variable z:
• stress space for Σ:

Z s B Ls Ω × Y ; V




×d 

S s B Ls Ω × Y ; Rdsym
.

• space for the micro displacement v:


1,s
V s B Ls Ω;Wper,⊥
(Y ; Rd ) .
∞ (Y ; Rd ) the space of C ∞ (Rd ; Rd ) functions which are Y -periodic,
For the latter, we denote by C per
1,s
∞ (Y ; Rd ) with
identified with their restriction on Y , and define Wper
(Y ; Rd ) to be the closure of C per
1,s
1,s
d
d
1,s
respect to the W (Y ; R ) norm. Further, W⊥ (Y ; R ) is the closed subspace of W (Y ; Rd ) consisting
of functions of mean 0, and
1,s
1,s
Wper,⊥
(Y ; Rd ) = Wper
(Y ; Rd ) ∩ W⊥1,s (Y ; Rd ).

We set the norm on V s to be
kv kV s B kv kLs (Ω×Y ;Rd ) + k∇sy v kLs (Ω×Y ;Rd ×d ) ,
with which V s becomes a Banach space and for the case s = 2 a Hilbert space with the obvious scalar
product.
Assumption 7.3 (Maximal monotone operator). The maximal monotone operator A from the evolution
2
law in (7.1d) is a set-valued map in the Hilbert space H = Z 2 , i.e., A: Z 2 → 2Z . It is assumed to satisfy
our standing assumptions from Section 2. Moreover, we assume that there is a sequence of operators
{An } from Z 2 to Z 2 satisfying Assumption 4.3.
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In Section 7.4 below, we will investigate the maximal monotone operator arising in the case of linear
kinematic hardening with von Mises yield condition and show how to construct the approximating
sequence of smooth operators for this particular case. With the above definitions at hand, we are now
in the position to define our precise notion of solutions to (7.1):
Definition 7.4 (Weak solutions). Let ` ∈ H 1 (0,T ; (U 2 )∗ ) and z 0 ∈ Z 2 . Then we say that a tuple
(u, v, z, Σ) ∈ H 1 (0,T ; U s ) × H 1 (0,T ; V 2 ) × H 1 (0,T ; Z 2 ) × H 1 (0,T ; S 2 )
is a solution of (7.1), if, for almost all t ∈ (0,T ), there holds
ˆ
(7.3a)
(π Σ(t))(x) · ∇xs φ(x) dx = h`(t), φi
Ω

(7.3b)

ˆ

(7.3c)
Ω×Y

Σ(t) = C(πr−1 ∇xs u(t) + ∇sy v(t) − Bz(t))

Σ(t, x, y) · ∇sy ψ (x, y) dy = 0

∀ φ ∈ U 2,
in S 2 ,
∀ψ ∈ V 2 ,

.

(7.3d)

z(t) ∈ A(B > Σ(t) − Bz(t))

in Z 2 ,

(7.3e)

z(0) = z 0

in Z 2 ,

×d ) is the average mapping from (7.2) and
where π : S 2 → L2 (Ω; Rdsym



×d
×d
πr−1 : L2 (Ω; Rdsym
) 3 ε 7→ Ω × Y 3 (x, y) 7→ ε(x) ∈ Rdsym
∈ S2.
In the following, we will frequently consider πr−1 in different domains and ranges, for simplicity denoted
by the same symbol.

7.2 reduction of the system
In the following, we reduce the system (7.3) to an equation in the internal variable z only and it will
turn out that this equation has exactly the form of our general equation (3.1). To this end we proceed
analog to [24, Chapter 4]. For this purpose, let us define the following operators:
Definition 7.5. Let s ∈ [1, ∞). Then we define
∇s(x, y ) : U s × V s → S s ,

∇s(x, y ) (u, v) B πr−1 ∇xs u + ∇sy v.

For its adjoint, we write
Div(x, y ) : S s → (U s )∗ × (V s )∗ ,
0

hDiv(x, y ) σ , (φ,ψ )i

B −h∇s(x, y ) ∗σ , (φ,ψ )i = −

ˆ
Ω×Y

σ (x, y) : (∇xs φ(x) + ∇sy ψ (x, y)) d(x, y).

With a slight abuse of notation, we denote these operators for different values of s always by the same
symbol.
Lemma 7.6. Let Assumption 7.1 be fulfilled. Then there is an index s̄ > 2 such that, for every s ∈ [s̄ 0, s̄]
0
0
and every (f, g) ∈ (U s )∗ × (V s )∗ , there exists a unique solution (u, v) ∈ U s × V s of


0
0
(7.4)
− Div(x, y ) C∇s(x, y ) (u, v) = (f, g) in (U s )∗ × (V s )∗
and there is a constant Cs > 0, independent of f and g, such that


k(u, v)kU s ×V s ≤ Cs kfk(U s 0 )∗ + kgk(V s 0 )∗ .
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Proof. The claim is equivalent to − Div(x, y ) C∇s(x, y ) being a topological isomorphism between U s × V s
and its dual space (U s )∗ × (V s )∗ for s ∈ [s̄ 0, s̄]. We start with the case s = 2. For this, the left hand side
of (7.4) gives rise to a bilinear form b on the Hilbert space U 2 × V 2 :




b (u, v), (φ,ψ ) B C∇s(x, y ) (u, v), ∇s(x, y ) (φ,ψ ) S 2
Clearly, b is bounded. Due to Poincaré’s inequality for functions with zero mean value, which implies
ˆ
ˆ


s
2
b (u, v), (u, v) ≥ c |Y |
|∇x u(x)| dx + c
|∇sy v(x, y)| 2 d(x, y)
Ω
Ω×Y

2
≥ C ku kH 1 (Ω;Rd ) + kv kV 2 (Ω×Y ;Rd )
∀ (u, v) ∈ U 2 × V 2 ,
D

it is also coercive so that the claim and isomorphism property for − Div(x, y ) C∇s(x, y ) for s = 2 follows
from the Lax-Milgram lemma.
We next extrapolate this isomorphism property to U s × V s for s around 2 using the fundamental
stability theorem by Šne˘iberg [36]. (See also the more accessible and extensive [3, Appendix A].) More
precisely, we show that the spaces U s × V s and their duals form complex interpolation scales in s.
Then the stability theorem shows that the set of scale parameters s such that − Div(x, y ) C∇s(x, y ) is a
topological isomorphism between U s × V s and its dual space is open. Since the set includes 2, as seen
above, this then implies the claim.
To establish the interpolation scale, it is enough to consider the primal case, since the dual interpolation scale is inherited from the primal one by duality properties of the complex interpolation
functor [38, Theorem 1.11.3]. So, we show that


U sθ × V sθ = U s0 × V s0 , U s1 × V s1 θ

for

1 1−θ θ
=
+
s
s0
s1

for all s 0 , s 1 ∈ (1, ∞) and θ ∈ (0, 1). It is moreover sufficient to consider each component in the
interpolation separately.
For the U s = WD1,s (Ω; Rd ) spaces, the interpolation scale property is well known by now in the
setting of Assumption 7.1 and even much more general ones; we refer to [6]. The result for V s is
1,s
proven by reducing the problem to the Wper,⊥
(Y ; Rd ) spaces and showing that these are complemented
subspaces of W 1,s (Y ; Rd ) and thus inherit the latter’s interpolation properties. This is done in the
appendix, Theorem b.3, and finishes the proof.

Remark 7.7. In general, one cannot expect s̄ to be significantly larger than 2, due to both the irregular
coefficient tensors and the mixed boundary conditions, see e.g. [18, 28, 34]. This issue will become
crucial in the discussion of second-order necessary optimality conditions in 7.3 below.
Now we are in the position to reduce (7.3) to an equation in the variable z only. For this purpose, we
need the following
Definition 7.8 (Q and R for the case of homogenized plasticity). Let s ∈ [s̄ 0, s̄] be given. By Lemma 7.6,
the solution operator associated with (7.4), denoted by

 −1
0
0
G B − Div(x, y ) C∇s(x, y ) : (U s )∗ × (V s )∗ → U s × V s ,
is well defined, linear and bounded. The components of G are abbreviated by
Gu B (1, 0) G : (U s )∗ × (V s )∗ → U s ,
0

0

Gv B (0, 1) G : (U s )∗ × (V s )∗ → V s .
0

0

Based on this solution operator, we moreover define
T : Z s 3 z 7→ B > C∇s(x, y ) G(− Div(x, y ) (CBz)) ∈ Z s .
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Now, we have everything at hand to define the mappings R and Q from our general equation (3.1) for
the special case of homogenized plasticity:
(7.5)

R : (U s )∗ 3 ` 7→ B > C∇s(x, y ) G(`, 0) ∈ Z s ,

(7.6)

Q : Z s 3 z 7→ (B > CB + B − T )z ∈ Z s .

0

Again, with a slight abuse of notation, we denote all of the above operators for different values of
s ∈ [s̄ 0, s̄] always by the same symbol.
The reason for defining the operators Q and R in the way we did in Definition 7.8 is the following:
Owing to Lemma 7.6, given z ∈ Z 2 , one can solve (7.3a)–(7.3c) for u, v, and Σ so that the tuple
(u, v, Σ) ∈ U 2 ×V 2 ×S 2 is uniquely determined by z. Even more, using the operators from Definition 7.8,
we see that the solution of (7.3a)–(7.3c) for given z is
(7.7)
(7.8)

(u, v) = G(− Div(x, y ) (CBz)) + (`, 0)),




Σ = C ∇s(x, y ) G − Div(x, y ) (CBz)) + (`, 0) − Bz .

Inserting the last equation in (7.3d) and employing the definition of Q and R in (7.6) and (7.5) then
yields
.
z ∈ A(B > Σ − Bz) = A(R` − Qz),
i.e., exactly an evolution equation of the general form in (3.1). This shows that the system (7.3) of
homogenized elastoplasticity can equivalently be rewritten as an abstract operator evolution equation
of the form (3.1).
For the differentiability properties needed in sections 5 and 6, a norm gap is required such that it is
no longer sufficient to consider just the Hilbert space H = Z 2 . In accordance with the definitions of R
and Q, we therefore define the spaces in the abstract setting in our concrete application problem as
follows:
Definition 7.9 (Spaces in case of homogenized plasticity). The spaces Y, Z, H , and W from Section 2
are set to
(7.9)

Y B Z s1 ,→ Z B Z s2 ,→ H = Z 2 ,→ W B Z s3

with s 1 ≥ s 2 ≥ 2 ≥ s 3 . The integrability indices s 1 , s 2 , and s 3 depend crucially on the differentiability
properties of the regularized version of A and will be specified for a concrete realization of A in
Section 7.4 below. Moreover, we choose
1,s 0

X B (U s1 )∗ = WD 1 (Ω; Rd )∗ C WD−1,s1 (Ω; Rd ).
0

Furthermore, the control space is given by
(7.10)

Xc B Lp (Ω; Rd ) × Lr (ΓN ; Rd )

1
with p > dds
+s 1

1
and r > (d −1)s
d .

Due to s 1 ≥ 2, Xc is reflexive and embeds compactly in X by Sobolev embedding and trace theorems.
Therefore, all our standing assumptions on the spaces in Section 2 are fulfilled.
Of course, elements in Xc are identified with those in X by
ˆ
ˆ
h(f , д), ui(U s1 )∗,U s1 B
f u dx +
д u ds, (f , д) ∈ Xc , u ∈ U s1 .
Ω

ΓN

In order to apply our general theory to the present setting, we need the following assumption on the
regularity of the linear equation (7.4). As we will see in subsections 7.3 and 7.4 below, this assumption
may become fairly restrictive, if one aims to establish second-order sufficient optimality conditions,
since, in this case, s 1 and the conjugate index s 30 may be rather large.
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Assumption 7.10 (Critical regularity condition). The index s̄ from Lemma 7.6 satisfies s̄ ≥ max{s 1 , s 30 },
where s 1 and s 3 are the integrability indices from (7.9).
Proposition 7.11. Under Assumption 7.10 and with the spaces defined in Definition 7.9, the operators R and
Q from (7.5) and (7.6), respectively, satisfy the standing assumptions from Section 2, that is, R is linear
0
and bounded from (U s1 )∗ to Z s1 and Q is a linear and bounded operator from Z s to Z s for all s ∈ [s 3 , s 1 ]
and, considered as an operator in Z 2 , coercive and self-adjoint.
Proof. The required mapping properties of Q and R directly follow from their construction in Definition 7.8 in combination with Lemma 7.6 and Assumption 7.10, respectively. It remains to show that
Q is coercive and self-adjoint. Since B is symmetric and coercive according to Assumption 7.1, it is
sufficient to prove that the operator B > CB − T : Z 2 → Z 2 is symmetric and positive. To prove the
symmetry, first observe that B > CB is symmetric by the symmetry of C. The symmetry of C moreover
implies that G : (U 2 )∗ × (V 2 )∗ → U 2 × V 2 , i.e., the solution operator of (7.4), is self-adjoint. Therefore,
the construction of T in Definition 7.8 implies for all z 1 , z 2 ∈ Z 2 that
(T z 1 , z 2 )Z 2 = h− Div(x, y ) (CBz 2 ), G(− Div(x, y ) (CBz 1 ))i
= hG(− Div(x, y ) (CBz 2 )), − Div(x, y ) (CBz 1 )i = (z 1 ,T z 2 )Z 2
so that T is also symmetric. To show the positivity of B > CB − T , let z ∈ Z 2 be arbitrary. To shorten
the notation, we abbreviate (uz , vz ) B G(− Div(x, y ) (CBz)). Then, by testing the equation for (uz , vz ),
i.e., (7.4) with (f, g) = − Div(x, y ) (CBz), with (−uz , −vz ), we arrive at


C(Bz − ∇s(x, y ) (uz , vz ), −∇s(x, y ) (uz , vz ) S 2 = 0.
Since, by construction, T z = B > C∇s(x, y ) (uz , vz ), the coercivity of C therefore implies




(B > CB − T )z, z Z 2 = C(Bz − ∇s(x, y ) (uz , vz ), Bz S 2


= C(Bz − ∇s(x, y ) (uz , vz ), Bz − ∇s(x, y ) (uz , vz ) S 2 ≥ 0.
As z was arbitrary, this proves the positivity.



We point out that the whole analysis in sections 3 and 4 is carried out in the Hilbert space H = Z 2 .
Therefore, for the mere existence and approximation results from these two sections, the critical
regularity condition in Assumption 7.10 is automatically fulfilled by setting s 1 = s 2 = s 3 = 2 (so that
Y = Z = W = H = Z 2 ). Note that, in this case, the Lax-Milgram lemma guarantees the assertion
of Assumption 7.10 without any further regularity assumptions, see the proof of Lemma 7.6. The
additional crucial regularity assumption only comes into play, when first- and second-order optimality
conditions are investigated, see Remark 2.1. In Section 7.4 below, we will elaborate in detail, where the
critical Assumption 7.10 is needed to ensure the required differentiability properties of the regularized
control-to-state map for the example of a specific yield condition.
We collect our findings so far in the following
Theorem 7.12 (Homogenized plasticity as abstract evolution VI). Under the Assumptions 7.1 and 7.3,
the system of homogenized elastoplasticty in its weak form in (7.3) is equivalent to an abstract operator
differential equation of the form
.

(7.11)

z ∈ A(R` − Qz),

z(0) = z 0 ,

with Q and R as defined in (7.5) and (7.6) in the following sense: If (u, v, z, Σ) solves (7.3), then z is
a solution (7.11), and vice versa, if z solves (7.11), then z together with (u, v) and Σ as defined in (7.7)
and (7.8), respectively, form a solution of (7.3).
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In addition, Q and R satisfy the standing assumptions from Section 2 (provided that the function spaces
are chosen according to Definition 7.9). Therefore, the existence and approximation results of Sections 3
and 4 hold for (7.11), in particular:
• For every ` ∈ U(z 0 , D(A)), there is a unique solution (u, v, z, Σ) of the weak system of homogenized
plasticity in (7.3), cf. Theorem 3.3.
• Optimal control problems governed by the weak system of homogenized elastoplasticity admit globally optimal solutions, provided that the standing assumptions on the objective are fulfilled, cf. Theorem 4.2.
• The approximation results of Theorem 4.5 and Corollary 4.6 apply in case of homogenized elastoplasticity.
Remark 7.13. Existence and uniqueness of solutions to (7.3) was already established in [33].
Remark 7.14. The example for objective functionals mentioned above, i.e.,
(7.12)

J (u, Σ, f , д) B α2

ˆ
Ω

|u(T ) − ud |

2

β
dx + 2

ˆ

γ

Ω

|(π Σ)(T ) − σd | 2 dx + 2 k(f , д)kH2 1 (0,T ;Xc )

×d ) and α, β ≥ 0, γ > 0, satisfies the standing assumptions on
with u D ∈ L2 (Ω; Rd ) and σ D ∈ L2 (Ω; Rdsym
the objective functional, as we will see in the following. In this case, the functional Ψ : (z, `) → R in the
general setting consists of the two integrals at end point T . Let us consider the first one containing the
displacement u. Since the latter is given by the first component of the solution operator of (7.7), which
maps H 1 (0,T ; Z 2 ) × H 1 (0,T ; Xc ) to H 1 (0,T ; U 2 ) ,→ C([0,T ]; L2 (Ω; Rd )), this integral is well defined.
(Note that the operators in (7.7) just act pointwise in time and the time regularity of z and ` carries
over to u and v.) Clearly, this solution operator is linear and bounded. In case of the second integral
involving Σ, one can argue completely analogously based on the solution operator of (7.8). Thus, Ψ is
convex and continuous, hence weakly lower semicontinuous, and in addition, bounded from below by
γ
zero. Moreover, the purely control part of the objective is given by Φ(f , д) = 2 k(f , д)kH2 1 (0,T ;X ) and
c
therefore clearly weakly lower semicontinuous and coercive as required. Thus all standing assumptions
are fulfilled as claimed. Of course, various other objective functionals are possible as well, such as
tracking type objectives over the whole space-time-cylinder, but to keep the discussion concise, we
just mention the example above.

7.3 optimality system
In the following section, we establish necessary and sufficient optimality conditions for the optimal
control of regularized homogenized elastoplasticity. To be more precise, we consider a single element
of the sequence of regularizations of the maximal monotone operator A from Assumption 7.3, which
we again denote by As , and apply the general theory from Section 5 and Section 6. In view of the norm
gap needed for the differentiability of As , we will consider As in different domains and ranges (denoted
by the same symbol) and assume that As maps Y = Z s1 to itself. Accordingly, we treat the regularized
version of the state equation (with As instead of A) in the same manner, i.e., with integrability index s 1
instead of 2, see (7.13) below. To keep the discussion concise, we moreover assume in all what follows
that s 1 ≥ 2 is such that p = r = 2 satisfy the conditions in (7.10). For d = dim(Ω) = 3, this implies
s 1 < 3 and, in case without boundary control, i.e., д ≡ 0, s 1 < 6 is sufficient. This will become important
in the discussion of second-order sufficient conditions, as we will see below. Motivated by (7.12), we
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consider an optimal control problem of the form

(7.13)





















min




















and

s.t.

J (u, Σ, f , д) B F 1 (u, Σ) + F 2 (u(T ), Σ(T ))

γ .
.
+ k f kL2 2 (0,T ;L2 (Ω;Rd )) + kдkL2 2 (0,T ;L2 (Γ ;Rd ))
N
2
1
2
d
2
d
(f , д) ∈ H (0,T ; L (Ω; R ) × L (ΓN ; R )),
(u, v, z, Σ) ∈ H 1 (0,T ; U s1 × V s1 × Z s1 × S s1 ),
− Div(x, y ) Σ = ((f , д), 0),


Σ = C ∇s(x, y ) (u, v) − Bz ,
.

z = As (B > Σ − Bz),

z(0) = z 0 ,

`(0) = 0.

Since, in many applications, displacement and stress on the macro level are of special interest, especially
at end time, we focus on objectives with this particular structure with continuously Fréchet differentiable
mappings
F 1 : L2 (0,T ; U 2 ) × L2 (0,T ; S 2 ) → R,

(7.14)

F 2 : U 2 × S 2 → R.

In order to apply our general theory, we not only have to reduce the state system to an equation of the
form (7.11), but also have to reduce the objective. For this purpose, let us denote the solution operators
of (7.7) and (7.8) by u : (`, z) 7→ u and S : (`, z) 7→ Σ. To shorten the notation, we will consider u and
0
0
S with different domains and ranges, e.g. u : (U s )∗ × Z s → U s and u : L2 (0,T ; (U s )∗ ) × L2 (0,T ; Z s ) →
L2 (0,T ; U s ) with s ∈ [s̄ 0, s̄] and analogously for S. Note again that the time regularity of z and ` directly
carries over to the time regularity of u and Σ. Given these operators, we define
Ψ1 : L2 (0,T ; Z 2 ) × L2 (0,T ; (U 2 )∗ ) → R, Ψ1 (z, `) B F 1 (u(z, `), S(z, `)),
Ψ2 : Z 2 × (U 2 )∗ → R,

Ψ2 (z, `) B F 2 (u(z, `), S(z, `)),

so that the objective in (7.13) becomes
γ

.

.

J (z, `) = Ψ1 (z, `) + Ψ2 (z(T ), `(T )) + 2 k( f , д)kL2 2 (0,T ;Xc ) ,

(7.15)

i.e., exactly an objective of the form in (5.15) and (5.18), respectively. Since u and S are linear and
bounded and F 1 and F 2 are assumed to be continuously Fréchet differentiable, the chain rule implies
the differentiability of Ψ1 and Ψ2 so that Assumption 5.1(i) is met, if we set s 3 = 2 so that W = Z 2 . To
apply the results of Section 5 in order to establish an optimality system for (7.13), we additionally need
that As satisfies Assumption 5.1(ii), which is ensured by the following
Assumption 7.15. We set s 2 = s 3 = 2, i.e., W = Z = Z 2 , and assume that As fulfills Assumption 5.1(ii)
with Y = Z s1 , s 1 ≥ 2, i.e., in particular that As is Fréchet differentiable from Z s1 to Z 2 .
In light of Lemma 7.6, Assumption 7.15 does not impose any restriction for practical realizations of
As , as we will see in Section 7.4 below. Given this assumption, Theorem 5.12 and Example 5.15 imply
for a locally optimal solution (`) = (f , д) with associated optimal internal variable z:
.

z(0) = z 0 ,


∂Ψ1
∂Ψ2
.
(7.16b)
φ(t) = QAs0 (R` − Qz)∗φ (t) +
(z, `)(t), φ(T ) = −
(z(T ), `(T )),
∂z
∂z

∗
∂Ψ1

2
∗ 0


 γ ∂t `(t) + R As R`(t) − Qz(t) φ(t) = ∂` (z, `)(t),

(7.16c)
∂Ψ2



`(0) = 0, ∂t `(T ) = −
(z(T ), `(T )).

∂`
Then, owing to the precise structure of R, Q, Ψ1 , and Ψ2 , this leads us to the following
(7.16a)

z(t) = As (R` − Qz)(t),

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Theorem 7.16 (KKT-system for optimal control of homogenized plasticity). Let Assumption 7.1 be satisfied and assume that As fulfills Assumption 7.15. Suppose moreover that the regularity condition in
Assumption 7.10 is satisfied, i.e., s̄ ≥ s 1 . Then, if (f , д) ∈ H 1 (0,T ; L2 (Ω)) × H 1 (0,T ; L2 (ΓN )) is locally optimal for (7.13) with associated state (u, v, z, Σ) ∈ H 1 (0,T ; U s1 )×H 1 (0,T ; V s1 )×H 1 (0,T ; Z s1 )×H 1 (0,T ; S s1 ),
then there exists an adjoint state
(w, q, φ, ϒ) ∈ H 1 (0,T ; U 2 ) × H 1 (0,T ; V 2 ) × H 1 (0,T ; Z 2 ) × H 1 (0,T ; S 2 ),
(wT , qT , ϒT ) ∈ U 2 × V 2 × S 2
such that the following optimality system is satisfied:
State equation:
− Div(x, y ) Σ = ((f , д), 0),

(7.17a)
(7.17b)



Σ = C ∇s(x, y ) (u, v) − Bz ,

(7.17c)

z = As (B > Σ − Bz),

.

z(0) = z 0 ,

Adjoint equation:
(7.17d)
(7.17e)
(7.17f)
(7.17g)
(7.17h)



0 >
∗ 

∂
∂u F 1 (u, Σ), 0 − Div(x, y ) CBAs (B Σ − Bz) φ ,


∂
ϒ = C ∇s(x, y ) (w, q) − ∂Σ
F 1 (u, Σ) ,
.
φ = (B > CB + B)As0 (B > Σ − Bz)∗φ + B > ϒ, φ(T ) = −B > ϒT ,


∂
− Div(x, y ) ϒT = ∂u
F 2 (u(T ), Σ(T )), 0 ,


∂
F 2 (u(T ), Σ(T )) ,
ϒT = C ∇s(x, y ) (wT , qT ) − ∂Σ
− Div(x, y ) ϒ =

Gradient equation:
(7.17i)

γ ∂t2 f + w = 0,

f (0) = 0,

γ ∂t f (T ) + wT = 0,

(7.17j)

γ ∂t2д + w = 0,

д(0) = 0,

γ ∂t д(T ) + wT = 0.

Remark 7.17. A passage to the limit w.r.t. the regularization in order to obtain an optimality system for
the original optimal control problem involving the maximal monotone operator A would of course
be of particular interest. The results of [43] however indicate that the optimality conditions obtained
in this way are in general rather weak. In [43], an optimal control problem governed by quasi-static
elastoplasticity (without homogenization) is considered, which provides substantial similarities to (7.13).
This system could also be treated by means of a reduction to the internal variable similar to our
procedure for (7.1). In [43] however, a time discretization followed by a regularization was employed for
the derivation of first-order optimality conditions. The reason for the comparatively weak optimality
conditions obtained for the original (non-smooth) problem is the poor regularity of the dual variables
in the limit, in particular the adjoint state. We expect a similar behavior in case of (7.17), when the
regularization is driven to zero. This however is subject to future research.
Next, we turn to second-order sufficient optimality conditions. Now, Ψ1 , Ψ2 , and As have to fulfill
Assumption 6.1. For this purpose, we require the following
Assumption 7.18. The mappings F 1 and F 2 from (7.14) are twice Fréchet differentiable. Moreover, As
satisfies Assumption 5.1 (ii) and Assumption 6.1(ii) and (iii) with W = H = Z 2 (i.e., s 3 = 2), Z = Z s2 ,
s 2 ≥ 2, and Y = Z s1 , s 1 ≥ s 2 .
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In contrast to Assumption 7.15, this assumption is very restrictive. If we assume that As arises as a
Nemyzki operator from a nonlinear function As : V → V (for simplicity denoted by the same symbol),
then the last condition in Assumption 6.1, i.e.,
kAs00(z)[z 1 , z 2 ]k W ≤ C kz 1 kZ s2 kz 2 kZ s2

(7.18)

∀ z ∈ Z s1 , z 1 , z 2 ∈ Z s2

with W = Z 2 , may only hold—even in case As00(z) ∈ L∞ (Ω × Y ; L(V, V))—provided that s 2 ≥ 2 s 3 = 4.
In order to have that As is Fréchet differentiable from Z s1 to Z s2 , we therefore need s 1 > 4 such that
Assumption 7.10 indeed becomes very restrictive, see Remark 7.7 and Remark 7.28 below. Moreover,
as described at the beginning of this subsection, if one sets r = 2, i.e., considers to boundary loads in
H 1 (0,T ; L2 (ΓN ; Rd )), then, in view of (7.10), s 1 < 3 has to hold (at least in three spatial dimensions) so
that our second-order analysis cannot be applied in case of boundary controls (at least if r = 2 and
d = 3). Therefore, we restrict to volume forces, i.e., controls in H 1 (0,T ; L2 (Ω; Rd )) in what follows.
Then, given all these assumptions, we can apply Theorem 6.7 and Corollary 6.9 to (7.13) to obtain
the following
Theorem 7.19 (Second-order sufficient conditions for optimal control of homogenized plasticity).
Let Assumptions 7.1 and 7.18 hold and suppose that s̄ ≥ s 1 so that Assumption 7.10 is fulfilled. Assume
moreover that f ∈ H 1 (0,T ; L2 (Ω; Rd )) together with its associated state (u, v, z, Σ) and an adjoint state
(w, q, φ, ϒ, wT , qT , ϒT ) satisfies the optimality system (7.17a)–(7.17i) (without (7.17j) because boundary
controls are omitted) and, in addition, that there exists δ > 0 such that
2
∂2
∂2
F (u, Σ)ηu2 + ∂Σ
2 F 1 (u, Σ)η Σ
∂u 2 1

.

2
2
2
∂
∂
+ ∂u
2 F 2 (u(T ), Σ(T ))ηu (T ) + ∂Σ2 F 2 (u(T ), Σ(T ))η Σ (T ) + γ khkL 2 (0,T ;L 2 (Ω;Rd ))
ˆ Tˆ


+
φ, As00(B > Σ − Bz)(B >η Σ − Bηz )2 V dx dt ≥ δ khkH2 1 (0,T ;L2 (Ω;Rd ))
2

2

0

Ω

holds for all h ∈ H 1 (0,T ; L2 (Ω; Rd )) with h(0) = 0, where (ηu , ηv , ηz , η Σ ) ∈ H 1 (0,T ; U 2 × V 2 × Z 2 × S 2 )
is the solution of the linearized state system associated with h, i.e.,
− Div(x, y ) η Σ = (h, 0),


η Σ = C ∇s(x, y ) (ηu , ηv ) − Bηz ,
.

ηz = As0 (B > Σ − Bz)(B >η Σ − Bηz ),

ηz (0) = 0.

Then f is a strict local minimizer of (7.13) and satisfies the quadratic growth condition (6.3).
Remark 7.20. As indicated above, Assumption 7.18 in combination with Assumption 7.10 is very
restrictive. One can however weaken these assumptions, if the objective provides certain properties.
Let us for instance consider an objective of the form
(7.19)

γ

.

J (u, f ) B F 3 (u) + 2 k f kL2 2 (0,T ;L2 (Ω;Rd ))

with a twice Fréchet differentiable functional
F 3 : H 1 (0,T ; L2 (Ω; Rd )) → R.
In this case, it is sufficient to choose s 3 such that u : (z, `) 7→ u maps W × Xc with W = Z s3 to
W 1,p (Ω; Rd ) ,→ L2 (Ω; Rd ), i.e., p ≥ 6/5 for d = 3. Moreover, as seen above, in order to have that the
Nemyzki operator As fulfills (7.18), we need s 1 > 2s 3 . Thus, we are tempted to choose s 3 as small as
possible. However, the crucial, limiting condition is the regularity assumption in Assumption 7.10
involving the conjugate exponent, i.e., s̄ ≥ max{s 1 , s 30 }, and this leads to the following equilibration
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of s 1 and s 3 in the case d = 3: s 1 > 3, s 3 = 3/2 (such that s 30 = 3). Then, in view of (7.7), u maps Z s3 to
W 1,s3 (Ω; Rd ), which is continuously embedded in L2 (Ω; Rd ) for d ≤ 3 as desired. In the next subsection,
we will present an example for a Nemyzki operator As fulfilling all assumptions for s 1 arbitrarily close
to 3. But nonetheless, s̄ > 3 in Assumption 7.10 is still a rather restrictive assumption and will not be
satisfied in general (depending on the regularity of C and the boundary of Ω). This shows that the
second-order analysis for problems of type (P) (resp. its regularized counterparts, to be precise) is in
general a delicate issue, mainly due to the quasi-linear structure of the state equation.

7.4 a concrete flow rule
In the following, we will discuss a concrete realization of the maximal monotone operator A and its
regularization, respectively, in order to demonstrate how the Assumptions 7.3, 7.15, and 7.18 can be
satisfied in practice and how restrictive they are, in particular Assumption 7.18.
We consider the case of linear kinematic hardening with von Mises yield condition, cf. [25] for a
detailed description of this model. In this case, the finite dimensional space for the internal variable is
×d and B : Rd ×d → Rd ×d is the identity so that Z 2 = S 2 . Moreover, A is the convex
given by V = Rdsym
sym
sym
subdifferential of the indicator functional I K of following set of admissible stresses
K B {τ ∈ S s : |τ D (x, y)| ≤ σ0

f.a.a. (x, y) ∈ Ω × Y },

×d and σ denotes the initial uni-axial yield
where τ D B τ − d1 tr(τ )I is the deviator of τ ∈ Rdsym
0
stress, a given material parameter. The domain of A = ∂I K is trivially K, which is closed and convex.
Furthermore, it is easily seen that A0 (τ ) = 0 for all τ ∈ D(A) = K so that all of our standing assumptions
are fulfilled in this case. Note moreover that A satisfies Assumption 3.9 so that the second approximation
result on the convergence of the optimal states in Corollary 4.6 applies in this case. For the Yosidaapproximation of ∂I K , one obtains
n
1
σ0 o
1
(7.20)
Aλ = (I − π K ) = max 0, 1 − D τ D ,
λ
λ
|τ |

cf. [26], where π K denotes the projection on K in Z 2 , i.e., π K (σ ) B arg minτ ∈K kτ −σ kZ2 2 . Herein, with
a slight abuse of notation, we denote the Nemyzki operator in L∞ (Ω × Y ) associated with the pointwise
maximum, i.e., R 3 r 7→ max{0, r } ∈ R, by the same symbol. In addition, we set max{0, 1 − σ0 /r } B 0,
if r = 0.
The precise form of Aλ in (7.20) immediately suggests the following regularization of the Yosida
approximation:

1
σ0 
Aλ,ϵ (τ ) B maxϵ 1 − D τ D ,
λ
|τ |
where maxϵ is a regularized version of the max-function, depending on a regularization parameter
ϵ > 0. To be more precise, maxϵ : L∞ (Ω × Y ) → L∞ (Ω × Y ) is the Nemyzki operator associated with a
real valued function (again denoted by the same symbol) with the following properties:
1. For every ϵ > 0, there holds maxϵ ∈ C 2 (R),
2. maxϵ (r ) = max{0, r } for |r | ≥ 21 and every 0 < ϵ ≤ 1/2,
3. | maxϵ (r ) − max{0, r }| ≤ O(ϵ) for all r ∈ R.
Example 7.21. An example for a function satisfying the above conditions is
(
max{0, r },
|r | ≥ ϵ,
maxϵ (r ) B
1
3
(r + ϵ) (3ϵ − r ), |r | < ϵ.
16ϵ 3
One easily verifies that maxε is twice continuously differentiable and that | maxε (r ) = max{0, r }| ≤ 163 ϵ.
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Lemma 7.22. Let {λn }n ∈N ⊂ R+ and {ϵn }n ∈N ⊂ R+ be null sequences satisfying



(7.21)
ϵn = o λn2 exp − T λkQn k ,
and define An B Aλn ,ϵn : Z 2 → Z 2 . Then, the sequence {An }n ∈N satisfies Assumption 4.3. Thus, Assumption 7.3 is fulfilled in this case so that the approximation results from Theorem 7.12 apply.
Proof. Based on our assumptions on maxϵ , we find for every τ ∈ Z 2 and all n ∈ N such that ϵn ≤ 1/2
that
kAn (τ ) − Aλn (τ )kZ2 2
ˆ
2



ϵ2
1
= 2
maxϵn 1 − |τσD0 | − max 0, 1 − |τσD0 | |τ D | 2 dx ≤ C n2 .
λn Ω
λn
The coupling of ϵn and λn in (7.21) then implies that (4.1) is fulfilled.



Remark 7.23. We point out that we neither claim that the coupling of λ and ϵ in (7.21) is optimal nor
that our regularization approach is the most efficient one for this specific flow rule.
Let us now fix n ∈ N and set λs B λn , maxs B maxϵn , and As B An . As before, we will denote the
Nemyzki operators generated by maxs and its derivatives by the same symbol. The following result
can be proven as in [27, Prop. 2.11] by using [22, Theorem 7]:
Lemma 7.24. Let s > r ≥ 1 be arbitrary. Then As is continuously Fréchet differentiable from Z s to Z r and
its directional derivative at τ ∈ Z s in direction h ∈ Z r is given by
As0 (τ )h =



1
σ0  σ0
1
σ0  D
D
D D
maxs0 1 − D
(τ
:
h
)τ
+
max
h .
1
−
s
λs
λs
|τ | |τ D | 3
|τ D |

As a consequence of this result, we obtain the following
Corollary 7.25. Assumption 7.15 is fulfilled by setting s 1 B s̄, where s̄ > 2 is the exponent, whose existence
is guaranteed by Lemma 7.6. Thus, in case of linear kinematic hardening with von Mises yield condition
and the regularization introduced above, the optimality condition in Theorem 7.16 are indeed necessary
for local optimality without any further assumptions (except our standing Assumption 7.1).
Proof. We have to verify Assumption 5.1(ii) for Y = Z s̄ and Z = H = Z 2 . The Fréchet differentiability
from Z s̄ to Z 2 follows from Lemma 7.24. Moreover, the (global) Lipschitz continuity of As in Z s̄ can
be deduced from the smoothness of maxs and the condition maxs (r ) = max{0, r } for all |r | ≥ 1/2.
The latter condition also guarantees that kAs0 (y)hkZ 2 ≤ C khkZ 2 for all y ∈ Z s̄ and all h ∈ Z 2 . This
completes the proof.

Furthermore, following the lines of [41] and [22, Theorem 9], one proves the following
Lemma 7.26. For every s > 2 and 1 ≥ r < s/2, As is twice Fréchet differentiable and its second derivative
at τ ∈ Z s in directions h 1 , h 2 ∈ Z r is given by
h
γ
γ 
 γ
As00(τ )[h 1 , h 2 ] = λ |τ D |3 maxs00 1 − |τ D | |τ D |3 (τ D : h 1D )(τ D : h 2D )τ D
s

γ 

+ maxs0 1 − |τ D | − |τ D3 |2 (τ D : h 1D )(τ D : h 2D )τ D + (h 1D : h 2D )τ D
i
+(τ D : h 1D )h 2D + (τ D : h 2D )h 1D .
Corollary 7.27. The conditions on As in Assumption 7.18 are satisfied, if the index s̄ from Lemma 7.6 fulfills
s̄ > 4.
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Proof. If we set s 1 = s̄ > 4,s 2 ∈]4, s 1 [, and s 3 = 2, then Lemma 7.26 implies the differentiability conditions
in Assumption 6.1(iii) with Y = Z s1 , Z = Z s2 , and W = Z s3 . The Lipschitz continuity of As0 from Z s1 to
L(Z s2 ) as well as the estimate kAs0 (y)w kZ 2 ≤ C kw kZ 2 follow from the condition maxs (r ) = max{0, r }
for all |r | ≥ 1/2. This condition also ensures that kAs00(y)[z 1 , z 2 ]kZ 2 ≤ C kz 1 kZ 4 kz 2 kZ 4 , which in turn
implies the last condition in Assumption 6.1(iii) thanks to s 2 > 4.

Remark 7.28. As indicated in Remark 7.7, the assumption s̄ > 4 is very restrictive. However, if W = Z 2 ,
then any Nemyzki operator is only twice Fréchet differentiable from Y to W, if Y = Z s with s > 4,
see e.g. [22] and the references therein. In light of this observation, the above regularization is rather
well-behaved. As explained in Remark 7.20, one can reduce the value of s 3 , if only the L2 -norm of
the displacement appears in the objective. However, one still needs s̄ > 3 in this case, which is not
guaranteed by Lemma 7.6 in general. But again, one can show that any s̄ > 3 is sufficient for our
concrete example, no matter how close it is to 3. This concrete realization of As thus allows for the
weakest possible regularity assumptions.

appendix a second derivative of the solution operator
Before we are in the position to show that S is twice Fréchet differentiable, we need the following
result on the Lipschitz continuity of the first derivative, which is also needed in the proof of Lemma 6.4.
Lemma a.1. Assume that Assumption 5.1(ii) and Assumption 6.1(ii) are fulfilled. Then Ss0 is Lipschitz
continuous from H 1 (0,T ; X) to L(H 1 (0,T ; X); H 1 (0,T ; Z)).
Proof of Proposition 6.3. Let `1 , `2 , h ∈ H 1 (0,T ; X) be arbitrary and abbreviate
ηi B Ss0 (`i )h,

zi B Ss (`i ),

and yi B R`i − Qzi ,

i = 1, 2.

Using the first Lipschitz-assumption in Assumption 6.1(ii), we deduce for almost all t ∈ [0,T ] that


.
.
kη 1 (t) − η 2 (t)k Z = k As0 (y1 (t)) − As0 (y2 (t)) (Rh(t) − Qη 1 (t)) + As0 (y2 (t))Q(η 1 (t) − η 2 (t))k Z


≤ C ky1 (t) − y2 (t)kY kRh(t) − Qη 1 (t)k Z + kη 1 (t) − η 2 (t)ik Z .
Gronwall’s inequality and the definition of y1 and y2 then yield
kη 1 − η 2 kH 1 (0,T ;Z) ≤ C kR(`1 − `2 ) − Q(z 1 − z 2 )kL2 (0,T ;Y ) kRh − Qη 1 kH 1 (0,T ;Z)
≤ C k`1 − `2 kL2 (0,T ;X) khkH 1 (0,T ;X) ,
where we used Lemma 5.3 and the estimate in Theorem 5.5.



Now, we are ready to prove that the solution operator is twice Fréchet-differentiable. The proof is
based on an estimate of the remainder term and thus similar to the one of Theorem 5.5.
Proof. The proof is similar to the one of Theorem 5.5. Let `, h 1 , h 2 ∈ H 1 (0,T ; X) be arbitrary and define
z B Ss (`), z 1 B Ss (` + h 1 ), ηi B Ss0 (`)hi ∈ H 1 (0,T ; Z), i ∈ {1, 2}, and η 1,2 B Ss0 (` + h 1 )h 2 .
We first address the existence of solutions to (6.1). We argue similarly to Lemma 5.4 and set
w : [0,T ] → W,

t 7→ As00(R`(t) − Qz(t))[Rh 1 (t) − Qη 1 (t), Rh 2 (t) − Qη 2 (t)].

From the estimate in Assumption 6.1(iii) it follows that
kw(t)k W ≤ C kRh 1 (t) − Qη 1 (t)k Z kRh 2 (t) − Qη 2 (t)k Z ,
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and, since the limit of Bochner measurable functions is Bochner measurable, we obtain w ∈ L2 (0,T ; W).
Since As0 (y) is assumed to be bounded in W by 6.1(ii), we can now follow the proof of Lemma 5.4 (with
W instead of Z) to deduce the existence of a unique solution ξ ∈ H 1 (0,T ; W) of (6.1). The (bi-)linearity
of the associated solution operator w.r.t. h 1 and h 2 is straightforward to see. For its continuity, we
calculate
.

k ξ (t)k W ≤ C kRh 1 (t) − Qη 1 (t)k Z kRh 2 (t) − Qη 2 (t)k Z + C kξ (t)k W
so that Gronwall’s inequality and the estimate in Theorem 5.5 give
kξ kH 1 (0,T ;W) ≤ C kh 1 kH 1 (0,T ;X) kh 2 kH 1 (0,T ;X) .
This shows also (6.2) (after having proved that ξ = Ss00(`)[h 1 , h 2 ]).
It only remains to prove the remainder term property. To this end, we define
y B R` − Qz,

ζ B Rh 1 − Q(z 1 − z).

Then, the equations for η 1,2 , η 2 , and ξ lead to
.


.
.
η 1,2 − η 2 − ξ = As0 (y + ζ ) − As0 (y) (Rh 2 − Qη 1,2 )

− As00(y)[Rh 1 − Qη 1 , Rh 2 − Qη 2 ] − As0 (y)Q(η 1,2 − η 2 − ξ )
= As00(y)[ζ , Rh 2 − Qη 1,2 ] + r 2 (y; ζ )(Rh 2 − Qη 1,2 )
− As00(y)[Rh 1 − Qη 1 , Rh 2 − Qη 2 ] − As0 (y)Q(η 1,2 − η 2 − ξ )
= As00(y)[ζ , Q(η 2 − η 1,2 )] − As00(y)[Q(z 1 − z − η 1 ), Rh 2 − Qη 2 ]
+ r 2 (y; ζ )(Rh 2 − Qη 1,2 ) − As0 (y)Q(η 1,2 − η 2 − ξ ),
where r 2 (y; ζ ) B As0 (y +ζ )−As0 (y)−As00(y)ζ ∈ L2 (0,T ; L(Z; W)) denotes the corresponding remainder
term. The estimate in Assumption 6.1(iii) thus implies
.

.

.

kη 1,2 (t) − η 2 (t) − ξ (t)k W
≤ C kζ (t))k Z kη 2 (t) − η 1,2 (t)k Z + kz 1 (t) − z(t) − η 1 (t)k Z kRh 2 (t) − Qη 2 (t)k Z
+ kr 2 (y(t), ζ (t))kL(Z;W) kRh 2 (t) − Qη 1,2 (t)k Z + kη 1,2 (t) − η 2 (t) − ξ (t)k W




for almost all t ∈ [0,T ] such that Gronwall’s inequality yields
kη 1,2 − η 2 − ξ kH 1 (0,T ;W)
≤ C kRh 1 − Q(z 1 − z)kL∞ (0,T ;Z) kη 2 − η 1,2 kL2 (0,T ;Z) + kz 1 − z − η 1 kL∞ (0,T ;Z) kRh 2 − Qη 2 kL2 (0,T ;Z)


+ kr 2 (y; ζ )kL2 (0,T ;L(Z;W)) kRh 2 − Qη 1,2 kH 1 (0,T ;Z)


≤ C kh 2 kH 1 (0,T ;X) kh 1 kH2 1 (0,T ;X) + kz 1 − z − η 1 kH 1 (0,T ;Z) + kr 2 (y; ζ )kL2 (0,T ;L(Z;W)) ,
where we used Lemma 5.3, Lemma a.1 and the estimate in Theorem 5.5. Denoting the solution operator
of (6.1) already by Ss00(`)[h 1 , h 2 ], we have thus shown
kSs0 (` + h 1 ) − Ss0 (`) − Ss00(`)h 1 kL(H 1 (0,T ;X);H 1 (0,T ;W))


≤ C kh 1 kH2 1 (0,T ;X) + kSs (` + h 1 ) − Ss (`) − Ss0 (`)h 1 kH 1 (0,T ;Z) + kr 2 (y; ζ )kL2 (0,T ;L(Z;W)) .
Therefore, thanks to the Fréchet differentiability of Ss : H 1 (0,T ; X) → H 1 (0,T ; Z), it only remains to
show that
(a.1)

H. Meinlschmidt, C. Meyer, S. Walther

kr 2 (y; ζ )kL2 (0,T ;L(Z;W))
→ 0,
kh 1 kH 1 (0,T ;X)
Optimal control of an abstract evolution variational . . .

J. Nonsmooth Anal. Optim. 1 (2020), 5800

page 36 of 41

as 0 , h 1 → 0 in H 1 (0,T ; X). To this end, we note that the embedding H 1 (0,T ; Y) ,→ C([0,T ]; Y) and
Lemma 5.3 yield for all t ∈ [0,T ]
(a.2)

kζ kH 1 (0,T ;Y)
kRh 1 − Q(z 1 − z)kH 1 (0,T ;Y)
kζ (t)k Y
≤C
=C
≤C
kh 1 kH 1 (0,T ;X)
kh 1 kH 1 (0,T ;X)
kh 1 kH 1 (0,T ;X)

Hence, thanks to the Fréchet differentiability of As0 : Y → L(Z; W), we have for almost all t ∈ [0,T ]
kr 2 (y; ζ )(t)kL(Z;W)
kr 2 (y; ζ )(t)kL(Z;W)
≤C
→0
kh 1 kH 1 (0,T ;X)
kζ (t)k Y
as 0 , h 1 → 0 in H 1 (0,T ; X). Furthermore, using the Lipschitz continuity of As0 : Y → L(Z; Z), the
estimate for As00 in Assumption 6.1(iii) and again (a.2), we deduce
kr 2 (y; ζ )(t)kL(Z;W)
kAs0 (y(t) + ζ (t)) − As0 (y(t)) − As00(y(t))ζ (t)kL(Z;W)
kζ (t)k Y
=
≤C
≤C
kh 1 kH 1 (0,T ;X)
kh 1 kH 1 (0,T ;X)
kh 1 kH 1 (0,T ;X)
for almost all t ∈ [0,T ]. The convergence in (a.1) now follows from Lebesgue’s dominated convergence
theorem.


appendix b interpolation for the V s spaces
1,s
We prove that the spaces V s = Ls (Ω;Wper,⊥
(Y ; Rd )) defined in Section 7 form a complex interpolation
scale in s. This result is a cornerstone in the proof of Lemma 7.6.
θ
Lemma b.1. Let θ ∈ (0, 1) and s 0 , s 1 ∈ (1, ∞) and set s1 = 1−θ
s 0 + s 1 . Then
h
i
1,s 0
1,s 1
1,s
(b.1)
Wper
(Y ; Rd ),Wper
(Y ; Rd ) = Wper
(Y ; Rd ).
θ

Proof. The proof relies on the complemented subspace interpolation theorem [38, Theorem 1.17.1]
which essentially says that one can transfer interpolation properties to complemented subspaces
provided there exists a common projection onto these subspaces on all involved spaces.
In this spirit, we first consider a larger regular domain Y # ⊃ Y which includes a finite open covering
1,r
1,r
of Y , and, for all 1 < r < ∞, identifyWper
(Y ; Rd ) isomorphically with the closed subspaceWper,Y
(Y # ; Rd )
1,r
of W 1,r (Y # ; Rd ) consisting of periodic extensions of Wper
(Y ; Rd )-functions. This is possible since the
1,r
d
periodic extension of a Wper (Y ; R ) function will preserve the Sobolev regularity [14, Proposition 3.50].
1,r
We next argue that there exists a projection P per mapping W 1,r (Y # ; Rd ) onto Wper,Y
(Y # ; Rd ). (We
will not give a detailed proof of this since the details are somewhat tedious and lengthy.) To this end,
we first wrap u ∈ W 1,r (Y # ; Rd ) around the torus T d in a smooth manner using a fixed smooth partition
of unity on T d derived from the open covering of Y , and then pull it back. One checks that this indeed
yields a function P peru which is periodic on Y . Moreover, P per is in fact a continuous linear operator
∞ (Y ; Rd ). This
on W 1,r (Y # ; Rd ), which in addition acts as the identity on the periodic extensions of C per
1,r
implies that P per is indeed the searched-for projection of W 1,r (Y # ; Rd ) onto Wper,Y
(Y # ; Rd ).
The complemented subspace interpolation theorem [38, Theorem 1.17.1] then allows to argue as
follows:
i
h
i
h
1,s 0
1,s 0
1,s 1
1,s 1
Wper
(Y ; Rd ),Wper
(Y ; Rd ) = Wper,Y
(Y # ; Rd ),Wper,Y
(Y # ; Rd )
θ
θ
h
i
1,max(s 0,s 1 ) #
1,max(s 0,s 1 ) #
1,s 0 #
d
d
1,s 1 #
d
= W (Y ; R ) ∩ Wper,Y
(Y ; R ),W (Y ; R ) ∩ Wper,Y
(Y ; Rd )
θ
h
i
1,max(s
,s
)
= W 1,s0 (Y # ; Rd ),W 1,s1 (Y # ; Rd ) ∩ Wper,Y 0 1 (Y # ; Rd )

=W
H. Meinlschmidt, C. Meyer, S. Walther

1,s

(Y

#

θ
1,max(s 0,s 1 ) #
1,s
1,s
; R ) ∩ Wper,Y
(Y ; Rd ) = Wper,Y
(Y # ; Rd ) = Wper
(Y ; Rd ).
d

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J. Nonsmooth Anal. Optim. 1 (2020), 5800

page 37 of 41

Here, the interpolation of W 1,r (Y # ; Rd ) is classical since we have assumed Y # to be regular. Overall,
this gives the claim.

θ
Lemma b.2. Let θ ∈ (0, 1) and s 0 , s 1 ∈ (1, ∞) and set s1 = 1−θ
s 0 + s 1 . Then

h
i
1,s 0
1,s 1
1,s
Wper,⊥
(Y ; RN ),Wper,⊥
(Y ; RN ) = Wper,⊥
(Y ; RN ).
θ

Proof. We again argue via the complemented subspace interpolation theorem. For every 1 < r < ∞,
the operator
P ⊥u := u −

Y

u dy

1,r
is a projection of W 1,r (Y ; Rd ) onto W⊥1,r (Y ; Rd ). Note that P ⊥ maps the closed subspace Wper
(Y ; Rd )
1,r
1,r
into itself, hence Wper,⊥ (Y ; Rd ) is a complemented subspace of Wper (Y ; Rd ) by means of the projection
P ⊥ . Using [38, Theorem 1.17.1] and Lemma b.1, we thus obtain

h
i
1,s 0
1,s 1
Wper,⊥
(Y ; Rd ),Wper,⊥
(Y ; Rd )
θ
h
i
1,s 0
1,s 1
d
= Wper (Y ; R ) ∩ W⊥1,max(s0,s1 ) (Y ; Rd ),Wper
(Y ; Rd ) ∩ W⊥1,max(s0,s1 ) (Y ; Rd )
θ
h
i
1,max(s 0,s 1 )
1,s 0
1,s 1
d
d
d
= Wper (Y ; R ),Wper (Y ; R ) ∩ W⊥
(Y ; R )

θ
1,s
1,s
= Wper (Y ; Rd ) ∩ W⊥1,max(s0,s1 ) (Y ; Rd ) = Wper,⊥
(Y ; Rd ),



as desired.
θ
Theorem b.3. Let θ ∈ (0, 1) and s 0 , s 1 ∈ (1, ∞) and set s1 = 1−θ
s 0 + s 1 . Then



Vs0 , Vs1 θ = Vs .
Proof. By [38, Thm. 1.18.4], we have
h



i


1,s 0
1,s 1
Vs0 , Vs1 θ = Ls0 Ω;Wper,⊥
(Y ; Rd )
(Y ; Rd ) , Ls1 Ω;Wper,⊥
θ
 1,s0
 

1,s 1
(Y ; Rd ),Wper,⊥
(Y ; Rd ) θ
= Ls Ω; Wper,⊥


1,s
= Ls Ω;Wper,⊥
(Y ; Rd ) = Vs ,
1,s
where the interpolation identity for Wper,⊥
(Y ; Rd ) is a consequence of Lemma b.2.



appendix c auxiliary results
Lemma c.1. Let M be a compact metric space and N a metric space. Furthermore, let {G n }n ∈N ⊂
C(M; N ), G ∈ C(M; N ) with G n (x) → G(x) for all x ∈ M and suppose that G n is uniformly Lipschitz continuous, that is, there exists a constant L such that
d N (G n (x), G n (y)) ≤ Ld M (x, y)
for all n ∈ N, x, y ∈ M.
Then G n → G in C(M; N ).
H. Meinlschmidt, C. Meyer, S. Walther

Optimal control of an abstract evolution variational . . .

J. Nonsmooth Anal. Optim. 1 (2020), 5800

page 38 of 41

Proof. We argue by contradiction. Assume that there exists ε > 0 and a strictly monotonically increasing
function n : N → N, such that for all k ∈ N there exists x k ∈ M with
ε ≤ d N (G n(k) (x k ), G(x k ))
for all k ∈ N. Since M is compact, we can extract a subsequence x k j of x k such that x k j → x in M,
thus
d N (G n(k j ) (x k j ), G(x k j )) ≤ d N (G n(k j ) (x k j ), G n(k j ) (x)) + d N (G n(k j ) (x), G(x k j ))
≤ Ld M (x k j , x) + d N (G n(k j ) (x), G(x k j )) → 0,


which gives the contradiction.

Lemma c.2. Let M be a compact metric space and N a metric space. Furthermore, let {G n }n ∈N ⊂
C(M; N ), G ∈ C(M; N ) with G n → G in C(M; N ). Define Un B G n (M) and U0 B G(M). Then
∞ U is compact.
the set U B ∪n=0
n
Proof. Let {yk }k ∈N ⊂ U . Since a finite union of compact sets is compact, we can assume that there
exists a subsequence {yk j }j ∈N and a strictly monotonically increasing function n : N → N, such that
yk j ∈ Un(j) . Then there exists a sequence {x j }j ∈N ⊂ M, with yk j = G n(j) (x j ). Because M is compact
we can select a subsequence, again denoted by x j , and a limit x ∈ M, such that x j → x, hence,
d N (yk j , G(x)) ≤ d N (yk j , G(x j )) + d N (G(x j ), G(x)) → 0,


thus the proof is complete.

references
[1] L. Adam and J. Outrata, On optimal control of a sweeping process coupled with an ordinary
differential equation, Discrete Contin. Dyn. Syst. Ser. B 19 (2014), 2709–2738, doi:10.3934/dcdsb.
2014.19.2709.
[2] C. E. Arroud and G. Colombo, A maximum principle for the controlled sweeping process, SetValued Var. Anal. 26 (2018), 607–629, doi:10.1007/s11228-017-0400-4.
[3] P. Auscher, S. Bortz, M. Egert, and O. Saari, Nonlocal self-improving properties: a functional
analytic approach, Tunisian J. Math. 1 (2019), 151–183, doi:10.2140/tunis.2019.1.151.
[4] V. Barbu, Optimal Control of Variational Inequalities, volume 100 of Research Notes in Mathematics, Pitman, Boston, 1984.
[5] V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press Boston,
1993.
[6] S. Bechtel and M. Egert, Interpolation theory for Sobolev functions with partially vanishing
trace on irregular open sets, Journal of Fourier Analysis and Applications (2019), doi:10.1007/
s00041-019-09681-1.
[7] H. Brézis, Opérateurs Maximaux Monotones, North-Holland, Amsterdam, 1973.
[8] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer Science
& Business Media, 2010, doi:10.1007/978-0-387-70914-7.
H. Meinlschmidt, C. Meyer, S. Walther

Optimal control of an abstract evolution variational . . .

J. Nonsmooth Anal. Optim. 1 (2020), 5800

page 39 of 41

[9] M. Brokate, Optimale Steuerung von gewöhnlichen Differentialgleichungen mit Nichtlinearitäten
vom Hysteresis-Typ, volume 35 of Methoden und Verfahren der Mathematischen Physik, Verlag
Peter D. Lang, Frankfurt am Main, 1987.
[10] M. Brokate and P. Krejčí, Optimal control of ODE systems involving a rate independent variational
inequality, Discrete Contin. Dyn. Syst. Ser. B 18 (2013), 331–348, doi:10.3934/dcdsb.2013.18.331.
[11] M. Brokate and P. Krejčí, Weak differentiability of scalar hysteresis operators, Discrete Contin.
Dyn. Syst. 35 (2015), 2405–2421, doi:10.3934/dcds.2015.35.2405.
[12] E. Casas and F. Tröltzsch, Second order necessary and sufficient conditions for optimization
problems and applications to control theory, SIAM Journal on Optimization 13 (2002), 406–431,
doi:10.1137/s1052623400367698.
[13] E. Casas and F. Tröltzsch, Second order optimality conditions and their role in PDE control, Jahresbericht der Deutschen Mathematiker-Vereinigung 117 (2015), 3–44, doi:10.1365/s13291-014-0109-3.
[14] D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford University Press, 1999.
[15] G. Colombo, R. Henrion, N. D. Hoang, and B. S. Mordukhovich, Discrete approximations of a
controlled sweeping process, Set-Valued Var. Anal. 23 (2015), 69–86, doi:10.1007/s11228-014-0299-y.
[16] G. Colombo, R. Henrion, N. D. Hoang, and B. S. Mordukhovich, Optimal control of the sweeping
process over polyhedral controlled sets, J. Differential Equations 260 (2016), 3397–3447, doi:10.
1016/j.jde.2015.10.039.
[17] G. Colombo and M. Palladino, The minimum time function for the controlled Moreau’s sweeping
process, SIAM J. Control Optim. 54 (2016), 2036–2062, doi:10.1137/15m1043364.
[18] J. Elschner, J. Rehberg, and G. Schmidt, Optimal regularity for elliptic transmission problems
including C 1 interfaces, Interfaces Free Bound. 9 (2007), 233–252, doi:10.4171/ifb/163.
[19] E. Emmrich, Gewöhnliche und Operator-Differentialgleichungen, Vieweg+Teubner Verlag, Wiesbaden, 2004, doi:10.1007/978-3-322-80240-8.
[20] H. Gajewski, K. Gröger, and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Mathematische Nachrichten 67 (1975), iv–iv, doi:10.1002/mana.19750672207.
[21] T. Geiger and D. Wachsmuth, Optimal control of an evolution equation with non-smooth dissipation, arXiv (2018), arXiv:1801.04077.
[22] H. Goldberg, W. Kampowsky, and F. Tröltzsch, On Nemytskij operators in Lp -spaces of abstract
functions, Mathematische Nachrichten 155 (1992), 127–140, doi:10.1002/mana.19921550110.
[23] K. Gröger, A W 1,p -estimate for solutions to mixed boundary value problems for second order
elliptic differential equations, Math. Ann. 283 (1989), 679–687, doi:10.1007/bf01442860.
[24] K. Gröger, Initial value problems for elastoplastic and elasto-viscoplastic systems, in Nonlinear
Analysis, Function Spaces and Applications, BSB B. G. Teubner Verlagsgesellschaft, 1979, 95–127,
http://eudml.org/doc/220893.
[25] W. Han and B. D. Reddy, Plasticity, volume 9 of Interdisciplinary Applied Mathematics, SpringerVerlag, New York, 1999, doi:10.1007/978-1-4614-5940-8.
H. Meinlschmidt, C. Meyer, S. Walther

Optimal control of an abstract evolution variational . . .

J. Nonsmooth Anal. Optim. 1 (2020), 5800

page 40 of 41

[26] R. Herzog and C. Meyer, Optimal control of static plasticity with linear kinematic hardening,
Journal of Applied Mathematics and Mechanics (ZAMM) 91 (2011), 777–794, doi:10.1002/zamm.
200900378.
[27] R. Herzog, C. Meyer, and G. Wachsmuth, C-stationarity for optimal control of static plasticity
with linear kinematic hardening, SIAM Journal on Control and Optimization 50 (2012), 3052–3082,
doi:10.1137/100809325.
[28] N. G. Meyers, An Lp -estimate for the gradient of solutions of second order elliptic divergence
equations, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 17 (1963), 189–206,
http://eudml.org/doc/83302.
[29] A. Mielke and T. Roubíček, Rate-Independent Systems, volume 193, Springer, New York, 2015,
doi:10.1007/978-1-4939-2706-7.
[30] J. J. Moreau, Problème d’évolution associé à un convexe mobile d’un espace Hilbertien, C. R.
Acad. Sci. Paris Sér. A-B 276 (1973), A791–A794.
[31] F. Rindler, Optimal control for nonconvex rate-independent evolution processes, SIAM J. Control
Optim. 47 (2008), 2773–2794, doi:10.1137/080718711.
[32] F. Rindler, Approximation of rate-independent optimal control problems, SIAM J. Numer. Anal.
47 (2009), 3884–3909, doi:10.1137/080744050.
[33] B. Schweizer and M. Veneroni, Homogenization of plasticity equations with two-scale convergence methods, Applicable Analysis 94 (2015), 375–398, doi:10.1080/00036811.2014.896992.
[34] E. Shamir, Regularization of mixed second-order elliptic problems, Israel J. Math. 6 (1968), 150–168,
doi:10.1007/bf02760180.
[35] R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations,
volume 49, American Mathematical Soc., 2013, doi:10.1090/surv/049.
[36] I. J. Šneı̆berg, Spectral properties of linear operators in interpolation families of Banach spaces,
Mat. Issled. 9 (1974), 214–229, 254–255.
[37] U. Stefanelli, D. Wachsmuth, and G. Wachsmuth, Optimal control of a rate-independent evolution
equation via viscous regularization, Discrete Contin. Dyn. Syst. Ser. S 10 (2017), 1467–1485, doi:
10.3934/dcdss.2017076.
[38] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, volume 18 of NorthHolland Mathematical Library, North-Holland Publishing Co., Amsterdam-New York, 1978.
[39] F. Tröltzsch, Optimale Steuerung partieller Differentialgleichungen, Springer, 2005, doi:10.1007/
978-3-322-96844-9.
[40] G. Wachsmuth, Optimal control of quasi-static plasticity with linear kinematic hardening, Part I:
Existence and discretization in time, SIAM J. Control Optim. 50 (2012), 2836–2861 + loose erratum,
doi:10.1137/110839187.
[41] G. Wachsmuth, Differentiability of implicit functions: Beyond the implicit function theorem,
Journal of Mathematical Analysis and Applications 414 (2014), 259–272, doi:10.1016/j.jmaa.2014.01.
007.
H. Meinlschmidt, C. Meyer, S. Walther

Optimal control of an abstract evolution variational . . .

J. Nonsmooth Anal. Optim. 1 (2020), 5800

page 41 of 41

[42] G. Wachsmuth, Optimal control of quasistatic plasticity with linear kinematic hardening II:
Regularization and differentiability, Z. Anal. Anwend. 34 (2015), 391–418, doi:10.4171/zaa/1546.
[43] G. Wachsmuth, Optimal control of quasistatic plasticity with linear kinematic hardening III:
Optimality conditions, Z. Anal. Anwend. 35 (2016), 81–118, doi:10.4171/zaa/1556.
[44] E. Zeidler, Nonlinear Functional Analysis and Its Applications II/A: Linear Monotone Operators,
Springer New York, 1990, doi:10.1007/978-1-4612-0985-0.
[45] E. Zeidler, Nonlinear Functional Analysis and its Applications III: Variational Methods and Optimization, Springer New York, 1985, doi:10.1007/978-1-4612-5020-3.

H. Meinlschmidt, C. Meyer, S. Walther

Optimal control of an abstract evolution variational . . .