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‎docs/src/2D-example.md

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@@ -6,14 +6,16 @@ We introduce the pseudo-Hamiltonian
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h(x,p,p^0,u) = p^0 x + p u.
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```
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For the sake of simplicity, we consider in this notebook only the normal case, and we fix $p^0 = -1$. According to the Pontryagin maximum principle, the maximizing control is given by $u(x,p) \to \mathrm{sign}(p)$. This function is non-differentiable, and may lead to numerical issues.
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For the sake of simplicity, we assume that the BC-extremals associated to the solution of the studied problem is normal, and so we fix $p^0 = -1$. According to the Pontryagin maximum principle, the maximizing control is given by $u(x,p) \to \mathrm{sign}(p)$. This function is non-differentiable, and may lead to numerical issues.
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Let's start by defining the problem.
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```@example main
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using OptimalControl # hide
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using Plots # hide
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using ForwardDiff # hide
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using DifferentialEquations # hide
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using MINPACK # hide
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using OptimalControl
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using Plots
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using ForwardDiff
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using DifferentialEquations
18+
using MINPACK
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t0 = 0 # initial time
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x0 = 0 # initial state
@@ -101,52 +103,14 @@ plot(sol) # plot
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Now, we want to provide $J_S$ to the solver, thanks to the $\texttt{ForwardDiff.jl}$ package. This Jacobian is computed with the variational equation, and leads to a false result in our case.
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Details. ( To be removed )
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Denoting $z = (x,p)$, we have
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```math
109-
\varphi(t_0, z_0, t_f) = z_0 + \int_{t_0}^{t_f} \vec H\big(\varphi(t_0, z_0, t)\big) \,\mathrm dt.
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```
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If we assume that $z_0 \to \varphi(t_0, z_0, t_f)$ is differentiable, we have
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```math
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\frac{\partial \varphi}{\partial z_0}(t_0, z_0, t_f)\cdot \delta z_0 = \delta z_0 + \int_{t_0}^{t_f} \vec H'\big(\varphi(t_0, z_0, t)\big)\cdot \left( \frac{\partial \varphi}{\partial z_0}(t_0, z_0, t) \cdot \delta z_0 \right) \,\mathrm dt,
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```
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and so, $z_0 \to \frac{\partial \varphi}{\partial z_0}(t_0, z_0, t_f)\cdot \delta z_0$ is solution of the variational equations
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```math
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\frac{\partial \delta z}{\partial t}(t) = \vec H'\big(\varphi(t_0, z_0, t_f)\big) \cdot \delta z(t), \qquad \delta z(t_0) = \delta z_0.
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```
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In the studied optimal control problem, we have
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```math
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\vec H(x,p) = (\mathrm{sign}(p), -1)
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```
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and so, we have $\vec H'(z) = 0_2$ almost everywhere, which implies
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```math
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\frac{\partial \varphi}{\partial z_0}(t_0, z_0, t_f) \cdot \delta z_0 = \mathrm{exp}\big((t_f-t_0) 0_2 \big)\cdot \delta z_0 = \delta z_0.
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```
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The Jacobian of the shooting function is then given by
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```math
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S'(p_0) = \pi \left( \frac{\partial \varphi}{\partial p_0}(t_0, x_0, p_0, t_f) \right) = \pi \left( \frac{\partial \varphi}{\partial z_0}(t_0, z_0, t_f) \cdot (0,1) \right) = \pi(0,1) = 0.
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```
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```@raw html
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<article class="docstring">
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<header>
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<a class="docstring-article-toggle-button fa-solid fa-chevron-right" href="javascript:;" title="Expand docstring"> </a>
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<span class="docstring-category">Details.</span>
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</header>
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<section style="display: none;">
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<p>Denoting <span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi><mo>=</mo><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">z = (x,p)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord mathnormal">p</span><span class="mclose">)</span></span></span></span></span></span>, we have </p>
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<p>Denoting <span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>z</mi><mn>0</mn></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>p</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">z_0 = (x_0,p_0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.5806em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span> the initial state costate couple, we have </p>
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<p class="math-container"><span><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>φ</mi><mo stretchy="false">(</mo><msub><mi>t</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>z</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>t</mi><mi>f</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>z</mi><mn>0</mn></msub><mo>+</mo><msubsup><mo>∫</mo><msub><mi>t</mi><mn>0</mn></msub><msub><mi>t</mi><mi>f</mi></msub></msubsup><mover accent="true"><mi>H</mi><mo>⃗</mo></mover><mo fence="false" stretchy="true" minsize="1.2em" maxsize="1.2em">(</mo><mi>φ</mi><mo stretchy="false">(</mo><msub><mi>t</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>z</mi><mn>0</mn></msub><mo separator="true">,</mo><mi>t</mi><mo stretchy="false">)</mo><mo fence="false" stretchy="true" minsize="1.2em" maxsize="1.2em">)</mo> <mi mathvariant="normal">d</mi><mi>t</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex"> \varphi(t_0, z_0, t_f) = z_0 + \int_{t_0}^{t_f} \vec H\big(\varphi(t_0, z_0, t)\big) \,\mathrm dt. </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1.0361em; vertical-align: -0.2861em;"></span><span class="mord mathnormal">φ</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3361em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1076em;">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.2861em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 0.7333em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.2222em;"></span></span><span class="base"><span class="strut" style="height: 2.5555em; vertical-align: -1.012em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right: 0.4445em; position: relative; top: -0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.5435em;"><span class="" style="top: -1.7881em; margin-left: -0.4445em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3173em;"><span class="" style="top: -2.357em; margin-left: 0em; margin-right: 0.0714em;"><span class="pstrut" style="height: 2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.143em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="" style="top: -3.8129em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3448em;"><span class="" style="top: -2.3488em; margin-left: 0em; margin-right: 0.0714em;"><span class="pstrut" style="height: 2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1076em;">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.2901em;"><span class=""></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 1.012em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.9663em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord mathnormal" style="margin-right: 0.0813em;">H</span></span><span class="" style="top: -3.2523em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.1799em;"><span class="overlay" style="height: 0.714em; width: 0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
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3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
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10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
@@ -199,7 +163,7 @@ println("ξ = ", ξ[1])
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println("JS(ξ) : ", JS(ξ)[1])
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```
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However, the solver $\texttt{hybrd1}$ uses rank 1 approximation to actualize the Jacobian insted of compute it at each iteration, which imply that it still converges to the solution even if the given Jacobian is completely false.
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However, the solver $\texttt{hybrd1}$ uses rank 1 approximations to actualize the Jacobian insted of compute it at each iteration, which imply that it still converges to the solution even if the given Jacobian is completely false.
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```@example main
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JS!(js, ξ) = (js[:] .= JS(ξ); nothing) # intermediate function
@@ -216,69 +180,7 @@ plt = plot(sol) # plot
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The goal is to provide the true Jacobian of $S$ by using the $\texttt{ForwardDiff}$ package, and so we need to indicate to the solver that the dynamic of the system change when $p = 0$.
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To understang why we need to give this information to the solver, see the following details.
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Details. (To be removed)
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The problem is that the Hamiltonian $H$ is not differentiable everywhere due to the maximizing control. This control is bang-bang ($u = 1$ and $u = -1$).
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Let now construct the two smooth Hamiltonians associated to these two controls
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```math
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H^+(x,p) = h(x,p,-1,1) = -x + p \qquad \text{and} \qquad H^-(x,p) = h(x,p,-1,-1) = -x - p.
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```
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Their associated vector fields are given by
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```math
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\vec H^+(x,p) = (1,1) \qquad \text{and} \qquad \vec H^-(x,p) = (-1, 1),
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```
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and their associated flow correspond to
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```math
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\varphi^+(t_0, z_0, t_f) = z_0 + \left( \begin{array}{c} 1 \\ 1 \end{array} \right) (t_f -t_0)
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\qquad \text{and} \qquad
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\varphi^-(t_0, z_0, t_f) = z_0 + \left( \begin{array}{c} -1 \\ \phantom{-} 1 \end{array} \right) (t_f -t_0).
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```
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If we assume that the optimal structure of the problem is negative then positive bangs, then the associated flow is defined by
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```math
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\varphi(t_0, z_0, t_f) = \varphi^+ \big( t_1(z_0), \varphi^-\big(t_0, z_0, t_1(z_0)\big), t_f \big),
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```
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with the following condition
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```math
254-
\pi_p \big( \varphi^-(t_0, z_0, t_1(z_0)) \big) = 0,
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```
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where $\pi_p(x,p) = p$ is the classical $p$-space projection. By devlopping this last condition, an explicit form of the function $t_1(\cdot)$ is given by
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```math
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t_1(x_0, p_0) = t_0 - p_0.
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```
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Finally, we have
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```math
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\begin{align*}
267-
\frac{\partial \varphi}{\partial z_0}
268-
&= \frac{\partial \varphi^+}{\partial t_0} \frac{\partial t_1}{\partial z_0} + \frac{\varphi^+}{\partial z_0} \left( \frac{\partial \varphi^-}{\partial z_0} + \frac{\partial \varphi^-}{\partial t_f} \frac{\partial t_1}{\partial z_0} \right) \\
269-
&= \left( \begin{array}{c} -1 \\ -1 \end{array} \right) \left( \begin{array}{cc}0 & -1 \end{array} \right) + \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right) \left[
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\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right) + \left( \begin{array}{c} -1 \\ \phantom - 1 \end{array} \right) \left( \begin{array}{cc}0 & -1 \end{array} \right)
271-
\right] \\
272-
&= \left( \begin{array}{cc} 0 & 1 \\ 0 & 1 \end{array} \right) + \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right) + \left( \begin{array}{cc} 0 & \phantom -1 \\ 0 & -1 \end{array} \right) \\
273-
&= \left( \begin{array}{cc} 1 & 2 \\ 0 & 1 \end{array} \right)
274-
\end{align*}
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```
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277-
and so, we have that
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279-
```math
280-
S'(p_0) = \pi \left( \frac{\partial \varphi}{\partial p_0}(t_0, x_0, p_0, t_f) \right) = \pi \left( \frac{\partial \varphi}{\partial z_0}(t_0, z_0, t_f) \cdot (0,1) \right) = \pi(2,1) = 2.
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```
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To understand why we need to give this information to the solver, see the following details.
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```@raw html
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<article class="docstring">

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