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+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Dynamic Optimization · DynOptInterface.jl</title><meta name="title" content="Dynamic Optimization · DynOptInterface.jl"/><meta property="og:title" content="Dynamic Optimization · DynOptInterface.jl"/><meta property="twitter:title" content="Dynamic Optimization · DynOptInterface.jl"/><meta name="description" content="Documentation for DynOptInterface.jl."/><meta property="og:description" content="Documentation for DynOptInterface.jl."/><meta property="twitter:description" content="Documentation for DynOptInterface.jl."/><meta property="og:url" content="https://JuDO-dev.github.io/DynOptInterface.jl/dynamic_optimization/"/><meta property="twitter:url" content="https://JuDO-dev.github.io/DynOptInterface.jl/dynamic_optimization/"/><link rel="canonical" href="https://JuDO-dev.github.io/DynOptInterface.jl/dynamic_optimization/"/><script data-outdated-warner 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src="../assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><a class="docs-logo" href="../"><img class="docs-light-only" src="../assets/logo.svg" alt="DynOptInterface.jl logo"/><img class="docs-dark-only" src="../assets/logo-dark.svg" alt="DynOptInterface.jl logo"/></a><div class="docs-package-name"><span class="docs-autofit"><a href="../">DynOptInterface.jl</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li><a class="tocitem" href="../">Home</a></li><li class="is-active"><a class="tocitem" href>Dynamic Optimization</a><ul class="internal"><li><a class="tocitem" href="#Phases-and-Dynamic-Variables"><span>Phases and Dynamic Variables</span></a></li><li><a class="tocitem" href="#Problem-Formulation"><span>Problem Formulation</span></a></li><li><a class="tocitem" href="#Dynamic-Functions"><span>Dynamic Functions</span></a></li><li><a class="tocitem" href="#Boundary-Functions"><span>Boundary Functions</span></a></li></ul></li><li><span class="tocitem">API Reference</span><ul><li><input class="collapse-toggle" id="menuitem-3-1" type="checkbox"/><label class="tocitem" for="menuitem-3-1"><span class="docs-label">Dynamic Functions</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../reference/dynamic_functions/abstraction/">Abstraction</a></li><li><a class="tocitem" href="../reference/dynamic_functions/phases/">Phases</a></li><li><a class="tocitem" href="../reference/dynamic_functions/dynamic_variables/">Dynamic Variables</a></li><li><a class="tocitem" href="../reference/dynamic_functions/expressions/">Expressions</a></li><li><a class="tocitem" href="../reference/dynamic_functions/derivatives/">Derivatives</a></li></ul></li><li><a class="tocitem" href="../reference/boundary_functions/">Boundary Functions</a></li><li><a class="tocitem" href="../reference/nonlinear_support/">Nonlinear Support</a></li><li><a class="tocitem" href="../reference/solutions/">Solutions</a></li></ul></li><li><a class="tocitem" href="../changelog/">Changelog</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>Dynamic Optimization</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Dynamic Optimization</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/JuDO-dev/DynOptInterface.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/JuDO-dev/DynOptInterface.jl/blob/dev/docs/src/dynamic_optimization.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Dynamic-Optimization"><a class="docs-heading-anchor" href="#Dynamic-Optimization">Dynamic Optimization</a><a id="Dynamic-Optimization-1"></a><a class="docs-heading-anchor-permalink" href="#Dynamic-Optimization" title="Permalink"></a></h1><h2 id="Phases-and-Dynamic-Variables"><a class="docs-heading-anchor" href="#Phases-and-Dynamic-Variables">Phases and Dynamic Variables</a><a id="Phases-and-Dynamic-Variables-1"></a><a class="docs-heading-anchor-permalink" href="#Phases-and-Dynamic-Variables" title="Permalink"></a></h2><p>Phases represent intervals <span>$t \in [t_0, t_f]$</span> on which dynamic variables <span>$t \mapsto \boldsymbol y(t)$</span> are defined.</p><p><img src="../assets/dynamic_variable.png" alt="dynamic variable"/></p><h2 id="Problem-Formulation"><a class="docs-heading-anchor" href="#Problem-Formulation">Problem Formulation</a><a id="Problem-Formulation-1"></a><a class="docs-heading-anchor-permalink" href="#Problem-Formulation" title="Permalink"></a></h2><p>MOI&#39;s <a href="https://jump.dev/MathOptInterface.jl/v1.31/manual/standard_form/#Standard-form-problem">Function-in-Set form problem</a> is extended as follows. Dynamic Optimization Problems deal with finding variables <span>$x \in \mathbb R^{n_x}$</span>, phase boundaries <span>$t_0^{(i)} \in \mathbb R$</span>, <span>$t_f^{(i)} \in \mathbb R$</span>, and dynamic variables <span>$\boldsymbol y^{(i)} : [t_0^{(i)}, t_f^{(i)}] \rightarrow \mathbb R^{n_y^{(i)}}$</span> that</p><p class="math-container">\[\begin{align*}
+\begin{array}{rl}
+    \text{minimize} \quad &amp;
+    m \big( \boldsymbol y(t_0), \boldsymbol y(t_f), t_0, t_f, x \big) +
+    \displaystyle{\sum_{i=1}^{n_p} \bigg[
+    \int_{t_0^{(i)}}^{t_f^{(i)}} \ell^{(i)} \big( \boldsymbol y^{(i)}(t), t, x \big) \textrm{d}t \bigg],}\\
+    %
+    \text{subject to} \quad &amp;
+    \begin{aligned}
+        f(x) &amp; \in \mathcal S,\\
+        %
+        d^{(i)}\big(\dot{\boldsymbol y}^{(i)}(t), \boldsymbol y^{(i)}(t), t, x) &amp; \in \mathcal D^{(i)},
+        \quad \forall t \in [t_0^{(i)}, t_f^{(i)}],
+        \quad \forall i \in \{1, 2, ..., n_p\},\\
+        b(\boldsymbol y(t_0), \boldsymbol y(t_f), t_0, t_f, x) &amp;\in \mathcal B,
+    \end{aligned}
+\end{array}
+\end{align*}\]</p><p>where:</p><ul><li><span>$f$</span> are <a href="https://jump.dev/MathOptInterface.jl/v1.31/reference/standard_form/#MathOptInterface.AbstractScalarFunction"><code>MOI.AbstractScalarFunction</code></a>s</li><li><span>$\ell$</span> and <span>$d$</span> are <a href="../reference/dynamic_functions/abstraction/#DynOptInterface.AbstractDynamicFunction"><code>AbstractDynamicFunction</code></a>s</li><li><span>$m$</span> and <span>$b$</span> are <a href="../reference/boundary_functions/#DynOptInterface.AbstractBoundaryFunction"><code>AbstractBoundaryFunction</code></a>s</li></ul><h2 id="Dynamic-Functions"><a class="docs-heading-anchor" href="#Dynamic-Functions">Dynamic Functions</a><a id="Dynamic-Functions-1"></a><a class="docs-heading-anchor-permalink" href="#Dynamic-Functions" title="Permalink"></a></h2><ul><li><a href="../reference/dynamic_functions/phases/#DynOptInterface.PhaseIndex"><code>PhaseIndex</code></a></li><li><a href="../reference/dynamic_functions/dynamic_variables/#DynOptInterface.DynamicVariableIndex"><code>DynamicVariableIndex</code></a></li><li><a href="../reference/dynamic_functions/expressions/#DynOptInterface.LinearDynamicFunction"><code>LinearDynamicFunction</code></a></li><li><a href="../reference/dynamic_functions/expressions/#DynOptInterface.PureQuadraticDynamicFunction"><code>PureQuadraticDynamicFunction</code></a></li><li><a href="../reference/dynamic_functions/expressions/#DynOptInterface.NonlinearDynamicFunction"><code>NonlinearDynamicFunction</code></a></li><li><a href="../reference/dynamic_functions/derivatives/#DynOptInterface.Derivative"><code>Derivative</code></a></li><li><a href="../reference/dynamic_functions/derivatives/#DynOptInterface.ExplicitDifferentialFunction"><code>ExplicitDifferentialFunction</code></a></li></ul><h2 id="Boundary-Functions"><a class="docs-heading-anchor" href="#Boundary-Functions">Boundary Functions</a><a id="Boundary-Functions-1"></a><a class="docs-heading-anchor-permalink" href="#Boundary-Functions" title="Permalink"></a></h2><ul><li><a href="../reference/boundary_functions/#DynOptInterface.Initial"><code>Initial</code></a></li><li><a href="../reference/boundary_functions/#DynOptInterface.Final"><code>Final</code></a></li><li><a href="../reference/boundary_functions/#Linkage"><code>Linkage</code></a></li><li><a href="../reference/boundary_functions/#DynOptInterface.NonlinearBoundaryFunction"><code>NonlinearBoundaryFunction</code></a></li><li><a href="../reference/boundary_functions/#DynOptInterface.Integral"><code>Integral</code></a></li><li><a href="../reference/boundary_functions/#DynOptInterface.MultiPhaseIntegral"><code>MultiPhaseIntegral</code></a></li><li><a href="../reference/boundary_functions/#DynOptInterface.Bolza"><code>Bolza</code></a></li></ul><script type="module">import mermaid from 'https://cdn.jsdelivr.net/npm/mermaid@10/dist/mermaid.esm.min.mjs';
+mermaid.initialize({
+    startOnLoad: true,
+    theme: "neutral"
+});
+</script></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../">« Home</a><a class="docs-footer-nextpage" href="../reference/dynamic_functions/abstraction/">Abstraction »</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="Saturday 31 August 2024 15:30">Saturday 31 August 2024</span>. Using Julia version 1.10.5.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>