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-<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Introduction · OptimalControl.jl</title><meta name="title" content="Introduction · OptimalControl.jl"/><meta property="og:title" content="Introduction · OptimalControl.jl"/><meta property="twitter:title" content="Introduction · OptimalControl.jl"/><meta name="description" content="Documentation for OptimalControl.jl."/><meta property="og:description" content="Documentation for OptimalControl.jl."/><meta property="twitter:description" content="Documentation for OptimalControl.jl."/><script data-outdated-warner src="assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" type="text/css"/><link 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href="assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit"><a href="index.html">OptimalControl.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 class="is-active"><a class="tocitem" href="index.html">Introduction</a></li><li><span class="tocitem">Tutorials</span><ul><li><a class="tocitem" href="tutorial-basic-example.html">Basic example</a></li><li><a class="tocitem" href="tutorial-basic-example-f.html">Basic example (functional version)</a></li><li><a class="tocitem" href="tutorial-double-integrator.html">Double integrator: time minimisation</a></li><li><a class="tocitem" href="tutorial-lqr-basic.html">LQR example</a></li><li><a class="tocitem" href="tutorial-goddard.html">Advanced example</a></li><li><a class="tocitem" href="tutorial-init.html">Initial guess</a></li><li><a class="tocitem" href="tutorial-plot.html">Plot a solution</a></li></ul></li><li><a class="tocitem" href="api.html">API</a></li><li><a class="tocitem" href="juliacon2023.html">JuliaCon2023</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="index.html">Introduction</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href="index.html">Introduction</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/control-toolbox/OptimalControl.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/control-toolbox/OptimalControl.jl/blob/main/docs/src/index.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="OptimalControl.jl"><a class="docs-heading-anchor" href="#OptimalControl.jl">OptimalControl.jl</a><a id="OptimalControl.jl-1"></a><a class="docs-heading-anchor-permalink" href="#OptimalControl.jl" title="Permalink"></a></h1><p>The <code>OptimalControl.jl</code> package aims to provide tools to solve optimal control problems by direct and indirect methods. It is part of the <a href="https://github.com/control-toolbox">control-toolbox ecosystem</a>:</p><img src="./assets/diagram.png" style="display: block; margin: 0 auto 20px auto;" width="320px"><div class="admonition is-info"><header class="admonition-header">Install and documentation</header><div class="admonition-body"><p>To install a package from the control-toolbox ecosystem,  please visit the <a href="https://github.com/control-toolbox#installation">installation page</a>. The documentation is accessible from the main menu.</p></div></div><p>An optimal control problem with fixed initial and final times can be described as minimising the cost functional</p><p class="math-container">\[g(x(t_0), x(t_f)) + \int_{t_0}^{t_f} f^{0}(t, x(t), u(t))~\mathrm{d}t\]</p><p>where the state <span>$x$</span> and the control <span>$u$</span> are functions subject, for <span>$t \in [t_0, t_f]$</span>, to the differential constraint</p><p class="math-container">\[   \dot{x}(t) = f(t, x(t), u(t))\]</p><p>and other constraints such as</p><p class="math-container">\[\begin{array}{llcll}
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Introduction · OptimalControl.jl</title><meta name="title" content="Introduction · OptimalControl.jl"/><meta property="og:title" content="Introduction · OptimalControl.jl"/><meta property="twitter:title" content="Introduction · OptimalControl.jl"/><meta name="description" content="Documentation for OptimalControl.jl."/><meta property="og:description" content="Documentation for OptimalControl.jl."/><meta property="twitter:description" content="Documentation for OptimalControl.jl."/><script data-outdated-warner src="assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL="."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="assets/documenter.js"></script><script src="search_index.js"></script><script src="siteinfo.js"></script><script src="../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit"><a href="index.html">OptimalControl.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 class="is-active"><a class="tocitem" href="index.html">Introduction</a></li><li><span class="tocitem">Tutorials</span><ul><li><a class="tocitem" href="tutorial-basic-example.html">Basic example</a></li><li><a class="tocitem" href="tutorial-basic-example-f.html">Basic example (functional version)</a></li><li><a class="tocitem" href="tutorial-double-integrator.html">Double integrator: time minimisation</a></li><li><a class="tocitem" href="tutorial-goddard.html">Goddard problem</a></li><li><a class="tocitem" href="tutorial-init.html">Initial guess</a></li><li><a class="tocitem" href="tutorial-lqr-basic.html">LQR example</a></li><li><a class="tocitem" href="tutorial-plot.html">Plot a solution</a></li></ul></li><li><a class="tocitem" href="api.html">API</a></li><li><a class="tocitem" href="juliacon2023.html">JuliaCon2023</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="index.html">Introduction</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href="index.html">Introduction</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/control-toolbox/OptimalControl.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/control-toolbox/OptimalControl.jl/blob/main/docs/src/index.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="OptimalControl.jl"><a class="docs-heading-anchor" href="#OptimalControl.jl">OptimalControl.jl</a><a id="OptimalControl.jl-1"></a><a class="docs-heading-anchor-permalink" href="#OptimalControl.jl" title="Permalink"></a></h1><p>The <code>OptimalControl.jl</code> package aims to provide tools to solve optimal control problems by direct and indirect methods. It is part of the <a href="https://github.com/control-toolbox">control-toolbox ecosystem</a>:</p><img src="./assets/diagram.png" style="display: block; margin: 0 auto 20px auto;" width="320px"><div class="admonition is-info"><header class="admonition-header">Install and documentation</header><div class="admonition-body"><p>To install a package from the control-toolbox ecosystem,  please visit the <a href="https://github.com/control-toolbox#installation">installation page</a>. The documentation is accessible from the main menu.</p></div></div><p>An optimal control problem with fixed initial and final times can be described as minimising the cost functional</p><p class="math-container">\[g(x(t_0), x(t_f)) + \int_{t_0}^{t_f} f^{0}(t, x(t), u(t))~\mathrm{d}t\]</p><p>where the state <span>$x$</span> and the control <span>$u$</span> are functions subject, for <span>$t \in [t_0, t_f]$</span>, to the differential constraint</p><p class="math-container">\[   \dot{x}(t) = f(t, x(t), u(t))\]</p><p>and other constraints such as</p><p class="math-container">\[\begin{array}{llcll}
 ~\xi_l  &amp;\le&amp; \xi(t, u(t))        &amp;\le&amp; \xi_u, \\
 \eta_l &amp;\le&amp; \eta(t, x(t))       &amp;\le&amp; \eta_u, \\
 \psi_l &amp;\le&amp; \psi(t, x(t), u(t)) &amp;\le&amp; \psi_u, \\
 \phi_l &amp;\le&amp; \phi(t_0, x(t_0), t_f, x(t_f)) &amp;\le&amp; \phi_u.
-\end{array}\]</p><p>See our tutorials to get started solving optimal control problems:</p><ul><li><a href="tutorial-basic-example.html#basic">Basic example</a>: a simple smooth optimal control problem solved by the direct method.</li><li><a href="tutorial-double-integrator.html#double-int">Double integrator</a>: the classical double integrator, minimum time. </li><li><a href="tutorial-goddard.html#goddard">Advanced example</a>: the <a href="https://en.wikipedia.org/w/index.php?title=Goddard_problem&amp;oldid=1091155283">Goddard problem</a> solved by direct and indirect methods.</li></ul></article><nav class="docs-footer"><a class="docs-footer-nextpage" href="tutorial-basic-example.html">Basic example »</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="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.1.0 on <span class="colophon-date" title="Wednesday 4 October 2023 20:03">Wednesday 4 October 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+\end{array}\]</p><p>See our tutorials to get started solving optimal control problems.</p></article><nav class="docs-footer"><a class="docs-footer-nextpage" href="tutorial-basic-example.html">Basic example »</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="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.1.0 on <span class="colophon-date" title="Thursday 5 October 2023 14:18">Thursday 5 October 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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