Merge pull request #18 from control-toolbox/final_version

new

.vscode/settings.json

@@ -0,0 +1,3 @@
+{
+    "julia.environmentPath": "c:\\Users\\Anas\\Desktop\\control-loss"
+}

Project.toml

@@ -1,4 +1,4 @@
 name = "ControlLoss"
 uuid = "6a3a0e7e-9c90-4c26-bc64-8dd630726a7b"
 authors = ["Olivier Cots <[email protected]>"]
-version = "0.1.4"
+version = "0.1.5"

docs/Project.toml

@@ -8,6 +8,7 @@ LaTeXStrings = "b964fa9f-0449-5b57-a5c2-d3ea65f4040f"
 LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
 NLPModelsIpopt = "f4238b75-b362-5c4c-b852-0801c9a21d71"
 NLsolve = "2774e3e8-f4cf-5e23-947b-6d7e65073b56"
+NonlinearSolve = "8913a72c-1f9b-4ce2-8d82-65094dcecaec"
 OptimalControl = "5f98b655-cc9a-415a-b60e-744165666948"
 OrdinaryDiffEq = "1dea7af3-3e70-54e6-95c3-0bf5283fa5ed"
 Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"

docs/make.jl

@@ -5,7 +5,7 @@ makedocs(
     sitename = "Control loss",
     format = Documenter.HTML(
         prettyurls = false, 
-        size_threshold_ignore = ["statement.md", "numerical.md","zermelo.md", "ho.md"],
+        size_threshold_ignore = ["statement.md", "numerical.md","zermelo1.md","zermelo2.md", "ho.md"],
         assets=[
             asset("https://control-toolbox.org/assets/css/documentation.css"),
             asset("https://control-toolbox.org/assets/js/documentation.js"),
@@ -15,7 +15,9 @@ makedocs(
         "Introduction"                       => "index.md",
         "Statement of the problem"           => "statement.md",
         "Numerical approach"                 => "numerical.md",
-        "Zermelo navigation problem"         => "zermelo.md",
+        "Zermelo navigation problem: Example 1"         => "zermelo1.md",
+        "Zermelo navigation problem: Example 2"         => "zermelo2.md",
+
         "Harmonic oscillator problem"        => "ho.md",
 
     ]

docs/src/ho.md

@@ -174,10 +174,13 @@ println(" p2(t1+) - p2(t1-) = ", jmp1)
 println(" p2(t2+) - p2(t2-) = ", jmp2)
 ```
 
+## Indirect Method 
+
+
 ```@example main
-using NLsolve
-using Animations
+using NonlinearSolve  
 using OrdinaryDiffEq
+using Animations
 ``` 
 
 ```@example main
@@ -260,33 +263,29 @@ nothing # hide
 ``` 
 
 ```@example main
-S(ξ) = shoot(ξ[1:2], ξ[3], ξ[4], ξ[5],ξ[6],ξ[7],ξ[8], ξ[9], ξ[10]);
+nle! =  (ξ, λ) -> shoot(ξ[1:2], ξ[3], ξ[4], ξ[5],ξ[6],ξ[7],ξ[8], ξ[9], ξ[10])
 ξ_guess = [p1(0) , p2(0), t1, t2, tstar , t3, a, jmp1, jmp2, t4]; # initial guess
-S(ξ_guess)  
+prob = NonlinearProblem(nle!, ξ_guess)
 ```
 
 ```@example main
-# Solve
+#solve
+indirect_sol = solve(prob; abstol=1e-8, reltol=1e-8, show_trace=Val(true))
+```
 
-indirect_sol = nlsolve(S, ξ_guess; xtol=1e-8, method=:trust_region, show_trace=true)
-println(indirect_sol)
 
+```@example main
 # Retrieves solution
-if indirect_sol.f_converged || indirect_sol.x_converged
-    pp0     = indirect_sol.zero[1:2]
-    tt1     = indirect_sol.zero[3]
-    tt2     = indirect_sol.zero[4]
-    ttstar  = indirect_sol.zero[5]
-    tt3     = indirect_sol.zero[6]
-    b11     = indirect_sol.zero[7]
-    jmp1    = indirect_sol.zero[8]
-    jmp2    = indirect_sol.zero[9]
-    T1      = indirect_sol.zero[10]
-
-else
-    error("Not converged")
-end
-nothing # hide
+     pp0     = indirect_sol[1:2]
+     tt1     = indirect_sol[3]
+     tt2     = indirect_sol[4]
+     ttstar  = indirect_sol[5]
+     tt3     = indirect_sol[6]
+     b11     = indirect_sol[7]
+     jmp1    = indirect_sol[8]
+     jmp2    = indirect_sol[9]
+     T1      = indirect_sol[10]
+ nothing # hide
 ```
 
 ```@example main

docs/src/numerical.md

@@ -34,15 +34,15 @@ for all $x\in \R^n$, where $\mathrm{d}_{j}: \R^n \to \R$ stands for the distance
 ## Description of the indirect method
 
 Recall that the direct method has captured the structure of the optimal pair $(x^*,u^*)$. In the indirect method, we address each arc separately. 
-We begin by defining the **flow** of the Hamiltonian associated with each arc. To accomplish this, we use the function `Flow` that can be found in the [`CTFlows.jl`](https://github.com/control-toolbox/CTFlows.jl) package. This latter allows to solve the Hamiltonian system over a given time interval from given initial values of the state and the adjoint vector. This function requires necessary libraries such as  \texttt{ForwardDiff} for calculating gradients and Jacobians and  `OrdinaryDiffEq` for solving ordinary differential equations. 
+We begin by defining the **flow** of the Hamiltonian associated with each arc. To accomplish this, we use the function `Flow` that can be found in the [CTFlows.jl](https://github.com/control-toolbox/CTFlows.jl) package. This latter allows to solve the Hamiltonian system over a given time interval from given initial values of the state and the adjoint vector. This function requires necessary libraries such as  ForwardDiff for calculating gradients and Jacobians and  OrdinaryDiffEq for solving ordinary differential equations. 
 
 In the setting of the present paper (including control regions and loss control regions), we distinguish between two types of Hamiltonian flows:
 
 - **Hamiltonian flows in control regions** We recall that the Hamiltonian  $H$ associated with the optimal control problem given above is defined by
 ```math
 H(x,u,p) := \langle  p, f(x,u)\rangle_{\R^n},
 ```
-for all $(x,u,p)\in \mathbb{R}^n \times \mathbb{R}^m \times \mathbb{R}^n$. Using the theorem given above and more specifically the Hamiltonian maximization condition, we obtain an expression of the control $u^*$ which can generate a sequence of arcs. Then it remains to define a \textit{pseudo-Hamiltonian}\footnote{the pseudo-Hamiltonian stands for the Hamiltonian flow associated with each arc.} associated with each arc. Finally we define the flow associated with each arc, which allows the resolution of the boundary value problem on a time interval satisfied by the pair $(x^*,p)$ with an initial condition.
+for all $(x,u,p)\in \mathbb{R}^n \times \mathbb{R}^m \times \mathbb{R}^n$. Using the theorem given above and more specifically the Hamiltonian maximization condition, we obtain an expression of the control $u^*$ which can generate a sequence of arcs. Then it remains to define a **pseudo-Hamiltonian** (the pseudo-Hamiltonian stands for the Hamiltonian flow associated with each arc). associated with each arc. Finally we define the flow associated with each arc, which allows the resolution of the boundary value problem on a time interval satisfied by the pair $(x^*,p)$ with an initial condition.
 
 - **Hamiltonian flows in loss control regions** Recall that, in loss control regions,  $u^*$ satisfies an averaged Hamiltonian gradient condition (see Theorem above). Here, the difficulty lies in the fact that this condition is given in an integral and implicit form. Therefore, to overcome this difficulty, we first introduce new states  $\lambda$ and  $y$. First, the state $\lambda$ comes from the augmentation technique (we refer to [^3] for more details) to handle the constant value  $u^*_k$, so it satisfies the dynamics  $\dot{\lambda}(t)=0$ and the initial condition  $\lambda(\tau^*_{k-1}) = u^*_k$. Second, the state $y$ satisfies $\dot{y}(t) = 0$ and $y(\tau^*_{k-1})=0$ (and thus $y=0$). Now we define the new Hamiltonian  $\tilde{H}$ as follows:
 ```math        

docs/src/statement.md

@@ -63,7 +63,7 @@ u \text{ is constant when } x \text{ is in a loss control region}.
 The PMP with loss control regions is based on some regularity assumptions made on the optimal pair of the optimal control problem with loss control regions given above at each of its crossing times. These hypotheses are made more precise in the next definition.
 
 !!! note "Definition (Regular solution to (CS))"
-    Following the notations introduced in Definition given above, a solution $(x,u) \in \mathrm{AC}([0,T],\R^n) \times \mathrm{L}^\infty([0,T],\R^m)$ to(CS), associated with a finite number $N \in \mathbb{N}^*$, a partition $\mathbb{T} = \{ \tau_k \}_{k=0,\ldots,N}$ and a switching sequence $\{j(1), \ldots, j(N)\}$, is said to be *regular* if the following conditions are both satisfied:
+    Following the notations introduced in Definition given above, a solution $(x,u) \in \mathrm{AC}([0,T],\R^n) \times \mathrm{L}^\infty([0,T],\R^m)$ to (CS), associated with a finite number $N \in \mathbb{N}^*$, a partition $\mathbb{T} = \{ \tau_k \}_{k=0,\ldots,N}$ and a switching sequence $\{j(1), \ldots, j(N)\}$, is said to be *regular* if the following conditions are both satisfied:
     - At each crossing time $\tau_k$, there exists a $\mathrm{C}^1$ function $F_k : \R^n \to \R$ such that 
     ```math
         \begin{equation}