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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 · Geometric preconditioner</title><meta name="title" content="Introduction · Geometric preconditioner"/><meta property="og:title" content="Introduction · Geometric preconditioner"/><meta property="twitter:title" content="Introduction · Geometric preconditioner"/><meta name="description" content="Documentation for Geometric preconditioner."/><meta property="og:description" content="Documentation for Geometric preconditioner."/><meta property="twitter:description" content="Documentation for Geometric preconditioner."/><script data-outdated-warner src="assets/warner.js"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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class="docs-package-name"><span class="docs-autofit"><a href="index.html">Geometric preconditioner</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><a class="tocitem" href="2D-example.html">2D example</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/preconditioning" 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/preconditioning/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="Geometric-preconditioner-for-indirect-shooting"><a class="docs-heading-anchor" href="#Geometric-preconditioner-for-indirect-shooting">Geometric preconditioner for indirect shooting</a><a id="Geometric-preconditioner-for-indirect-shooting-1"></a><a class="docs-heading-anchor-permalink" href="#Geometric-preconditioner-for-indirect-shooting" title="Permalink"></a></h1><p>blabla</p></article><nav class="docs-footer"><a class="docs-footer-nextpage" href="2D-example.html">2D 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 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Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Introduction · Geometric preconditioner</title><meta name="title" content="Introduction · Geometric preconditioner"/><meta property="og:title" content="Introduction · Geometric preconditioner"/><meta property="twitter:title" content="Introduction · Geometric preconditioner"/><meta name="description" content="Documentation for Geometric preconditioner."/><meta property="og:description" content="Documentation for Geometric preconditioner."/><meta property="twitter:description" content="Documentation for Geometric preconditioner."/><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/catppuccin-macchiato.css" data-theme-name="catppuccin-macchiato"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/catppuccin-frappe.css" data-theme-name="catppuccin-frappe"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/catppuccin-latte.css" data-theme-name="catppuccin-latte"/><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><link href="https://control-toolbox.org/assets/css/documentation.css" rel="stylesheet" type="text/css"/><script src="https://control-toolbox.org/assets/js/documentation.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit"><a href="index.html">Geometric preconditioner</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><a class="tocitem" href="2D-example.html">Indirect shooting</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/preconditioning" 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/preconditioning/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="Optimal-control-problem"><a class="docs-heading-anchor" href="#Optimal-control-problem">Optimal control problem</a><a id="Optimal-control-problem-1"></a><a class="docs-heading-anchor-permalink" href="#Optimal-control-problem" title="Permalink"></a></h1><p>We consider the following optimal control problem </p><p class="math-container">\[    \left\{ \begin{array}{ll}
+    \displaystyle \min_{x,u} \int_{t_0}^{t_f} x(t) ~\mathrm dt \\[1em]
+    \text{s.c.}~\dot x(t) = u(t), &amp; t\in [t_0, t_f]~\mathrm{a.e.}, \\[0.5em]
+    \phantom{\mathrm{s.c.}~} u(t) \in [-1,1], &amp; t\in [t_0, t_f], \\[0.5em]
+    \phantom{\mathrm{s.c.}~} x(t_0) = x_0, \quad x(t_f) = x_f,
+    \end{array} \right.\]</p><p>with <span>$x_0$</span>, <span>$t_0$</span>, <span>$x_f$</span> and <span>$t_f$</span> fixed. This problem is simple, and can be analytically solve without the use of numerical method. However, the goal is to solve this problem by indirect shooting.  </p></article><nav class="docs-footer"><a class="docs-footer-nextpage" href="2D-example.html">Indirect shooting »</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="Monday 12 August 2024 16:51">Monday 12 August 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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