Add tutorials on free times

docs/make.jl

@@ -17,7 +17,7 @@ cp(
 repo_url = "github.com/control-toolbox/Tutorials.jl"
 
 makedocs(;
-    draft=false, # if draft is true, then the julia code from .md is not executed
+    draft=true, # if draft is true, then the julia code from .md is not executed
     # to disable the draft mode in a specific markdown file, use the following:
     # ```@meta
     # Draft = false
@@ -38,6 +38,12 @@ makedocs(;
         "Tutorials and Advanced Features" => [
             "Discrete continuation" => "tutorial-continuation.md",
             "Discretisation methods" => "tutorial-discretisation.md",
+            "Free times" => [
+                "Final time" => "tutorial-free-times-final.md",
+                "Initial time" => "tutorial-free-times-initial.md",
+                "Final and initial times" => "tutorial-free-times-final-initial.md",
+                "Orbital transfer min time" => "tutorial-free-times-orbital.md",
+            ],
             "NLP manipulations" => "tutorial-nlp.md",
             "Indirect simple shooting" => "tutorial-iss.md",
             "Goddard: direct, indirect" => "tutorial-goddard.md",

docs/src/tutorial-continuation.md

@@ -1,3 +1,4 @@
+
 # Discrete continuation
 
 By using the warm start option, it is easy to implement a basic discrete continuation method, in which a sequence of problems is solved by using each solution as the initial guess for the next problem.
@@ -53,7 +54,7 @@ Then we perform the continuation with a simple *for* loop, using each solution t
 init = ()
 data = DataFrame(T=Float64[], Objective=Float64[], Iterations=Int[])
 for T ∈ range(1, 2, length=5)
-    ocp = problem(T) 
+    ocp = problem(T)
     sol = solve(ocp; init=init, display=false)
     global init = sol
     push!(data, (T=T, Objective=objective(sol), Iterations=iterations(sol)))

docs/src/tutorial-free-times-final-initial.md

@@ -0,0 +1,54 @@
+```@meta
+Draft = false
+```
+
+# [Optimal control problem with free initial and free final times](@id tutorial-free-times-final-initial)
+
+In this tutorial, we explore optimal control problems with free initial time `t₀` and final time `t_f`. 
+
+
+## Here are the required packages for the tutorial:
+
+```@example both_time
+using OptimalControl
+using NLPModelsIpopt
+using BenchmarkTools
+using DataFrames
+using Plots
+using Printf
+```
+## Definition of the problem
+
+```@example both_time
+
+@def ocp begin
+    v=(t0, tf) ∈ R², variable
+    t ∈ [t0, tf], time
+    x ∈ R², state
+    u ∈ R, control
+    -1 ≤ u(t) ≤ 1
+    x(t0) == [0, 0]
+    x(tf) == [1, 0]
+    0.05 ≤ t0 ≤ 10
+    0.05 ≤ tf ≤ 10
+    0.01 ≤ tf - t0 ≤ Inf
+    ẋ(t) == [x₂(t), u(t)]
+    t0 → max
+end
+
+nothing # hide
+```
+
+#  Direct resolution with both final and inital times being free:
+
+
+We now solve the problem using a direct method, with automatic treatment of the free initial time.
+
+```@example both_time
+sol = solve(ocp; grid_size=100)
+```
+And plot the solution.
+
+```@example both_time
+plot(sol; label="direct", size=(800, 800))
+```