build based on 368f586

dev/.documenter-siteinfo.json

@@ -1 +1 @@
-{"documenter":{"julia_version":"1.10.4","generation_timestamp":"2024-07-16T21:47:15","documenter_version":"1.5.0"}}
+{"documenter":{"julia_version":"1.10.4","generation_timestamp":"2024-07-16T22:10:53","documenter_version":"1.5.0"}}

dev/application-surface-revolution.html

@@ -126,7 +126,7 @@
 plot!(plt_u; xlabel="t", ylabel="u(t,p₀)", legend=false, ylims=(-2.5, 5))
 plot!(plt_phase; xlabel="x(t,p₀)", ylabel="p(t,p₀)", legend=false, xlims=(0, 2), ylims=(-1, 2))
 
-plot(plt_x, plt_p, plt_u, plt_phase; layout=(2, 2), size=(800, 600))</code></pre><img src="application-surface-revolution-b177a6a7.svg" alt="Example block output"/><p>Here, the shooting equation given by </p><p class="math-container">\[    S({p₀}) = \pi(z(t_f,x_0,{p₀})) - x_f = 0,\]</p><p>with <span>$\pi(x, p) = x$</span>, has two solutions: <span>$p₀ = -0.9851$</span> and <span>$p₀ = 0.5126$</span>.</p><pre><code class="language-julia hljs">π((x, p)) = x
+plot(plt_x, plt_p, plt_u, plt_phase; layout=(2, 2), size=(800, 600))</code></pre><img src="application-surface-revolution-0babaa0e.svg" alt="Example block output"/><p>Here, the shooting equation given by </p><p class="math-container">\[    S({p₀}) = \pi(z(t_f,x_0,{p₀})) - x_f = 0,\]</p><p>with <span>$\pi(x, p) = x$</span>, has two solutions: <span>$p₀ = -0.9851$</span> and <span>$p₀ = 0.5126$</span>.</p><pre><code class="language-julia hljs">π((x, p)) = x
 
 # Shooting function
 S(p0) = (π ∘ ocp_flow)(t0, x0, p0, tf) - xf
@@ -171,7 +171,7 @@
 plot!(plt2_u; xlabel=&quot;t&quot;, ylabel=&quot;u(t,p₀)&quot;, legend=false, ylims=(-6, 5))
 plot!(plt2_phase; xlabel=&quot;x(t,p₀)&quot;, ylabel=&quot;p(t,p₀)&quot;, legend=false, xlims=(0, 2.5), ylims=(-1, 5))
 
-plot(plt2_x, plt2_p, plt2_u, plt2_phase; layout=(2, 2), size=(800, 600))</code></pre><img src="application-surface-revolution-02ce5e03.svg" alt="Example block output"/><p>Now, we can compute the conjugate points along the two extremals. We have to compute the flow <span>$\delta z(t, p₀)$</span> of the Jacobi equation  with the initial condition <span>$\delta z(0) = (0, 1)$</span>. This is given solving</p><p class="math-container">\[    \delta z(t, p₀) = \dfrac{\partial}{\partial p₀}z(t, p₀).\]</p><p>Note that to compute the conjugate points, we only need the first component:</p><p class="math-container">\[    \delta z(t, p₀)_1.\]</p><pre><code class="language-julia hljs">using ForwardDiff
+plot(plt2_x, plt2_p, plt2_u, plt2_phase; layout=(2, 2), size=(800, 600))</code></pre><img src="application-surface-revolution-edcd8615.svg" alt="Example block output"/><p>Now, we can compute the conjugate points along the two extremals. We have to compute the flow <span>$\delta z(t, p₀)$</span> of the Jacobi equation  with the initial condition <span>$\delta z(0) = (0, 1)$</span>. This is given solving</p><p class="math-container">\[    \delta z(t, p₀) = \dfrac{\partial}{\partial p₀}z(t, p₀).\]</p><p>Note that to compute the conjugate points, we only need the first component:</p><p class="math-container">\[    \delta z(t, p₀)_1.\]</p><pre><code class="language-julia hljs">using ForwardDiff
 
 function jacobi_flow(t, p0)
     x(t, p0) = (π ∘ ocp_flow)(t0, x0, p0, t)
@@ -200,11 +200,11 @@
 
 #
 plt_conj = plot(plt_conj1, plt_conj2;
-    layout=(1, 2), size=(800, 300), leftmargin=25px, bottommargin=15px)</code></pre><img src="application-surface-revolution-9ec02a3e.svg" alt="Example block output"/><p>We compute the first conjugate point along the first extremal and add it to the plot.</p><pre><code class="language-julia hljs">tau0 = Roots.find_zero(tau -&gt; jacobi_flow(tau, sol1_p0), (0.4, 0.6))
+    layout=(1, 2), size=(800, 300), leftmargin=25px, bottommargin=15px)</code></pre><img src="application-surface-revolution-fbad296f.svg" alt="Example block output"/><p>We compute the first conjugate point along the first extremal and add it to the plot.</p><pre><code class="language-julia hljs">tau0 = Roots.find_zero(tau -&gt; jacobi_flow(tau, sol1_p0), (0.4, 0.6))
 
 println(&quot;For p0 = &quot;, sol1_p0, &quot; tau_0 = &quot;, tau0)
 
-plot!(plt_conj[1], [tau0], [jacobi_flow(tau0, sol1_p0)]; seriestype=:scatter)</code></pre><img src="application-surface-revolution-b40ef26c.svg" alt="Example block output"/><p>To conclude on this example, we compute the conjugate locus by using a path following algorithm. Define <span>$F(\tau,p₀) = \delta x(\tau,p₀)$</span> and suppose that the partial  derivative <span>$\partial_\tau F(\tau,p₀)$</span> is invertible, then, by the implicit function  theorem the conjugate time is a function of <span>$p₀$</span>. So, since here <span>$p₀\in\R$</span>, we can  compute them by solving the initial value problem for  <span>$p₀ \in [\alpha, \beta]$</span>:</p><p class="math-container">\[    \dot{\tau}(p₀) = -\dfrac{\partial F}{\partial \tau}(\tau(p₀),p₀)^{-1}\, 
+plot!(plt_conj[1], [tau0], [jacobi_flow(tau0, sol1_p0)]; seriestype=:scatter)</code></pre><img src="application-surface-revolution-59ce891b.svg" alt="Example block output"/><p>To conclude on this example, we compute the conjugate locus by using a path following algorithm. Define <span>$F(\tau,p₀) = \delta x(\tau,p₀)$</span> and suppose that the partial  derivative <span>$\partial_\tau F(\tau,p₀)$</span> is invertible, then, by the implicit function  theorem the conjugate time is a function of <span>$p₀$</span>. So, since here <span>$p₀\in\R$</span>, we can  compute them by solving the initial value problem for  <span>$p₀ \in [\alpha, \beta]$</span>:</p><p class="math-container">\[    \dot{\tau}(p₀) = -\dfrac{\partial F}{\partial \tau}(\tau(p₀),p₀)^{-1}\, 
     \dfrac{\partial F}{\partial p₀}(\tau(p₀),p₀), \quad
     \tau(\alpha) = \tau_0.\]</p><p>For the numerical experiment, we set <span>$\alpha = -0.9995$</span>, <span>$\beta = -0.5$</span>.</p><pre><code class="language-julia hljs">function conjugate_times_rhs_path(tau, p0)
     dF = ForwardDiff.gradient(y -&gt; jacobi_flow(y...), [tau, p0])
@@ -238,4 +238,4 @@
 
 #
 plot(plt_x, plt_conj_times;
-    layout=(1,2), legend=false, size=(800,300), leftmargin=25px, bottommargin=15px)</code></pre><img src="application-surface-revolution-944edafc.svg" alt="Example block output"/><section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-1"><a class="tag is-link" href="#citeref-1">1</a>H. Schättler &amp; U. Ledzewicz, <em>Geometric optimal control: theory, methods and examples</em>, vol~<strong>38</strong> of <em>Interdisciplinary applied mathematics</em>, Springer Science &amp; Business Media, New York (2012), xiv+640.</li><li class="footnote" id="footnote-2"><a class="tag is-link" href="#citeref-2">2</a>D. Liberzon, <em>Calculus ov Variations and Optimal Control Theory</em>, Princeton University Press (2012).</li></ul></section></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="index.html">« Introduction</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 1.5.0 on <span class="colophon-date" title="Tuesday 16 July 2024 21:47">Tuesday 16 July 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+    layout=(1,2), legend=false, size=(800,300), leftmargin=25px, bottommargin=15px)</code></pre><img src="application-surface-revolution-e70d2beb.svg" alt="Example block output"/><section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-1"><a class="tag is-link" href="#citeref-1">1</a>H. Schättler &amp; U. Ledzewicz, <em>Geometric optimal control: theory, methods and examples</em>, vol~<strong>38</strong> of <em>Interdisciplinary applied mathematics</em>, Springer Science &amp; Business Media, New York (2012), xiv+640.</li><li class="footnote" id="footnote-2"><a class="tag is-link" href="#citeref-2">2</a>D. Liberzon, <em>Calculus ov Variations and Optimal Control Theory</em>, Princeton University Press (2012).</li></ul></section></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="index.html">« Introduction</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 1.5.0 on <span class="colophon-date" title="Tuesday 16 July 2024 22:10">Tuesday 16 July 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>

dev/index.html

@@ -20,4 +20,4 @@
   [1dea7af3] OrdinaryDiffEq v6.85.0
   [91a5bcdd] Plots v1.40.5
   [f2b01f46] Roots v2.1.5
-  [fd094767] Suppressor v0.2.7</code></pre></article><nav class="docs-footer"><a class="docs-footer-nextpage" href="application-surface-revolution.html">Catenoid solution »</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 1.5.0 on <span class="colophon-date" title="Tuesday 16 July 2024 21:47">Tuesday 16 July 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+  [fd094767] Suppressor v0.2.7</code></pre></article><nav class="docs-footer"><a class="docs-footer-nextpage" href="application-surface-revolution.html">Catenoid solution »</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 1.5.0 on <span class="colophon-date" title="Tuesday 16 July 2024 22:10">Tuesday 16 July 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>