build based on 9f89fba

dev/.documenter-siteinfo.json

@@ -1 +1 @@
-{"documenter":{"julia_version":"1.10.4","generation_timestamp":"2024-08-22T11:30:09","documenter_version":"1.6.0"}}
+{"documenter":{"julia_version":"1.10.4","generation_timestamp":"2024-08-22T11:33:29","documenter_version":"1.6.0"}}

dev/ho.html

@@ -28,7 +28,7 @@
     include(&quot;smooth.jl&quot;);</code></pre><pre><code class="language-julia hljs">a  = 0.0
 ε1 = 0.02
 fNC(x) = fNC_unboundedminus(x,a,ε1)
-plot(fNC,-1., 1, label=&quot;fNC&quot;)</code></pre><img src="ho-be21d698.svg" alt="Example block output"/><pre><code class="language-julia hljs">  @def ocp begin
+plot(fNC,-1., 1, label=&quot;fNC&quot;)</code></pre><img src="ho-8f0ad150.svg" alt="Example block output"/><pre><code class="language-julia hljs">  @def ocp begin
 
     ε = 1e-3
 
@@ -306,17 +306,17 @@
 Number of equality constraint Jacobian evaluations   = 186
 Number of inequality constraint Jacobian evaluations = 0
 Number of Lagrangian Hessian evaluations             = 185
-Total seconds in IPOPT                               = 7.253
+Total seconds in IPOPT                               = 6.665
 
-EXIT: Optimal Solution Found.</code></pre><pre><code class="language-julia hljs">plot(sol; layout=:group, size=(800, 300))</code></pre><img src="ho-5b1019a1.svg" alt="Example block output"/><pre><code class="language-julia hljs">tt    = sol.times
+EXIT: Optimal Solution Found.</code></pre><pre><code class="language-julia hljs">plot(sol; layout=:group, size=(800, 300))</code></pre><img src="ho-6a303619.svg" alt="Example block output"/><pre><code class="language-julia hljs">tt    = sol.times
 tf    = tt[end]
 x1(t) = sol.state(t)[1]
 x2(t) = sol.state(t)[2]
 λ(t)  = sol.state(t)[3]
 u(t)  = sol.control(t)[1];</code></pre><pre><code class="language-julia hljs"># Plot the optimal trajectory
 plot(x1, x2, 0, tf, label=&quot;optimal trajectory&quot;, color=&quot;blue&quot;, linewidth=2)
-plot!([-4, 4], [0, 0], color=:black, label=false, linewidth=2)</code></pre><img src="ho-99f66e26.svg" alt="Example block output"/><pre><code class="language-julia hljs">plot(tt, u, label=&quot;optimal control&quot;, color=&quot;red&quot;, linewidth=2)
-plot!(tt, λ, label=&quot;state λ&quot;, color=&quot;green&quot;, linewidth=2)</code></pre><img src="ho-90a68694.svg" alt="Example block output"/><pre><code class="language-julia hljs"># Find the crossing times based on conditions for x1
+plot!([-4, 4], [0, 0], color=:black, label=false, linewidth=2)</code></pre><img src="ho-cc80fb5d.svg" alt="Example block output"/><pre><code class="language-julia hljs">plot(tt, u, label=&quot;optimal control&quot;, color=&quot;red&quot;, linewidth=2)
+plot!(tt, λ, label=&quot;state λ&quot;, color=&quot;green&quot;, linewidth=2)</code></pre><img src="ho-21a95f7b.svg" alt="Example block output"/><pre><code class="language-julia hljs"># Find the crossing times based on conditions for x1
 t1_index = findfirst(t -&gt; x2(t) ≤ 0, tt)
 t2_index = nothing
 t3_index = nothing
@@ -350,4 +350,4 @@
 println(&quot;Fourth crossing time: &quot;, t44)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">First crossing time: 0.8657506162825221
 Second crossing time: 4.0161209144217
 Third crossing time: 6.517178250348986
-Fourth crossing time: 9.619451292028025</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="zermelo.html">« Zermelo navigation problem</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.6.0 on <span class="colophon-date" title="Thursday 22 August 2024 11:30">Thursday 22 August 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+Fourth crossing time: 9.619451292028025</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="zermelo.html">« Zermelo navigation problem</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.6.0 on <span class="colophon-date" title="Thursday 22 August 2024 11:33">Thursday 22 August 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>

dev/index.html

@@ -14,4 +14,4 @@
   [1dea7af3] OrdinaryDiffEq v6.87.0
   [91a5bcdd] Plots v1.40.5
   [71555da5] Reel v1.3.2
-  [37e2e46d] LinearAlgebra</code></pre><section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-1"><a class="tag is-link" href="#citeref-1">1</a>L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mishchenko, <em>The Mathematical Theory of Optimal Processes</em>, A Pergamon Press Book. The Macmillan Co., New York, 1964.</li><li class="footnote" id="footnote-2"><a class="tag is-link" href="#citeref-2">2</a>T. Bayen, A. Bouali, L. Bourdin &amp; O. Cots, <em>Loss control regions in optimal control problems</em>, Journal of Differential Equations, <strong>405</strong> (2024), 359-397.</li><li class="footnote" id="footnote-3"><a class="tag is-link" href="#citeref-3">3</a>P. Bettiol, L. Bourdin, <em>Pontryagin Maximum Principle for State Constrained Optimal Sampled-Data Control Problems on Time Scales</em>, ESAIM Control Optim. Calc. Var., <strong>27</strong> (2021) 51.</li><li class="footnote" id="footnote-4"><a class="tag is-link" href="#citeref-4">4</a>L. Bourdin, G. Dhar, <em>Optimal Sampled-Data Controls with Running Inequality State Constraints: Pontryagin Maximum Principle and Bouncing Trajectory Phenomenon</em>, Mathematical Programming, <strong>191</strong> (2022) 907–951.</li><li class="footnote" id="footnote-5"><a class="tag is-link" href="#citeref-5">5</a>T. Haberkorn, E. Trélat, <em>Convergence Results for Smooth Regularizations of Hybrid Nonlinear Optimal Control Problems</em>, SIAM Journal on Control and Optimization, <strong>49</strong> (2011) 1498–1522.</li></ul></section></article><nav class="docs-footer"><a class="docs-footer-nextpage" href="statement.html">Statement of the problem »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.6.0 on <span class="colophon-date" title="Thursday 22 August 2024 11:30">Thursday 22 August 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+  [37e2e46d] LinearAlgebra</code></pre><section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-1"><a class="tag is-link" href="#citeref-1">1</a>L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mishchenko, <em>The Mathematical Theory of Optimal Processes</em>, A Pergamon Press Book. The Macmillan Co., New York, 1964.</li><li class="footnote" id="footnote-2"><a class="tag is-link" href="#citeref-2">2</a>T. Bayen, A. Bouali, L. Bourdin &amp; O. Cots, <em>Loss control regions in optimal control problems</em>, Journal of Differential Equations, <strong>405</strong> (2024), 359-397.</li><li class="footnote" id="footnote-3"><a class="tag is-link" href="#citeref-3">3</a>P. Bettiol, L. Bourdin, <em>Pontryagin Maximum Principle for State Constrained Optimal Sampled-Data Control Problems on Time Scales</em>, ESAIM Control Optim. Calc. Var., <strong>27</strong> (2021) 51.</li><li class="footnote" id="footnote-4"><a class="tag is-link" href="#citeref-4">4</a>L. Bourdin, G. Dhar, <em>Optimal Sampled-Data Controls with Running Inequality State Constraints: Pontryagin Maximum Principle and Bouncing Trajectory Phenomenon</em>, Mathematical Programming, <strong>191</strong> (2022) 907–951.</li><li class="footnote" id="footnote-5"><a class="tag is-link" href="#citeref-5">5</a>T. Haberkorn, E. Trélat, <em>Convergence Results for Smooth Regularizations of Hybrid Nonlinear Optimal Control Problems</em>, SIAM Journal on Control and Optimization, <strong>49</strong> (2011) 1498–1522.</li></ul></section></article><nav class="docs-footer"><a class="docs-footer-nextpage" href="statement.html">Statement of the problem »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.6.0 on <span class="colophon-date" title="Thursday 22 August 2024 11:33">Thursday 22 August 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>

dev/numerical.html

@@ -17,4 +17,4 @@
     \end{array}
     \right.\]</p><p>Second, since  <span>$u_k^*$</span> is assumed to be an interior value to  <span>$\mathrm{U}$</span>, we get that</p><p class="math-container">\[\begin{equation*}
     \int_{\tau^*_{k-1}}^{\tau^*_k} \nabla_{u} H(x^*(t),u^*_k,p(t)) \, dt = {p}_{y}(\tau^*_k)=0.
-\end{equation*}\]</p><p>Hence there is no need to compute an integral in order to take into account the averaged Hamiltonian gradient condition. </p><section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-1"><a class="tag is-link" href="#citeref-1">1</a>S. Aronna, F. Bonnans, P. Martinon, <em>A shooting algorithm for optimal control problems with singular arcs</em>, Journal of Optimization Theory and Applications, <strong>158</strong>, pp. 419–459, Springer, 2013.</li><li class="footnote" id="footnote-2"><a class="tag is-link" href="#citeref-2">2</a>J.F. Bonnans, <em>The shooting approach to optimal control problems</em>, IFAC Proceedings Volumes, <strong>46</strong>, no. 11, pp. 281–292, Elsevier, 2013.</li><li class="footnote" id="footnote-3"><a class="tag is-link" href="#citeref-3">3</a>T. Bayen, A. Bouali, L. Bourdin &amp; O. Cots, <em>Loss control regions in optimal control problems</em>, Journal of Differential Equations, <strong>405</strong> (2024), 359-397.</li></ul></section></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="statement.html">« Statement of the problem</a><a class="docs-footer-nextpage" href="zermelo.html">Zermelo navigation problem »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.6.0 on <span class="colophon-date" title="Thursday 22 August 2024 11:30">Thursday 22 August 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+\end{equation*}\]</p><p>Hence there is no need to compute an integral in order to take into account the averaged Hamiltonian gradient condition. </p><section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-1"><a class="tag is-link" href="#citeref-1">1</a>S. Aronna, F. Bonnans, P. Martinon, <em>A shooting algorithm for optimal control problems with singular arcs</em>, Journal of Optimization Theory and Applications, <strong>158</strong>, pp. 419–459, Springer, 2013.</li><li class="footnote" id="footnote-2"><a class="tag is-link" href="#citeref-2">2</a>J.F. Bonnans, <em>The shooting approach to optimal control problems</em>, IFAC Proceedings Volumes, <strong>46</strong>, no. 11, pp. 281–292, Elsevier, 2013.</li><li class="footnote" id="footnote-3"><a class="tag is-link" href="#citeref-3">3</a>T. Bayen, A. Bouali, L. Bourdin &amp; O. Cots, <em>Loss control regions in optimal control problems</em>, Journal of Differential Equations, <strong>405</strong> (2024), 359-397.</li></ul></section></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="statement.html">« Statement of the problem</a><a class="docs-footer-nextpage" href="zermelo.html">Zermelo navigation problem »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.6.0 on <span class="colophon-date" title="Thursday 22 August 2024 11:33">Thursday 22 August 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>