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+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Components of a Neural Lyapunov Problem · NeuralLyapunov.jl</title><meta name="title" content="Components of a Neural Lyapunov Problem · NeuralLyapunov.jl"/><meta property="og:title" content="Components of a Neural Lyapunov Problem · NeuralLyapunov.jl"/><meta property="twitter:title" content="Components of a Neural Lyapunov Problem · NeuralLyapunov.jl"/><meta name="description" content="Documentation for NeuralLyapunov.jl."/><meta property="og:description" content="Documentation for NeuralLyapunov.jl."/><meta property="twitter:description" content="Documentation for NeuralLyapunov.jl."/><meta property="og:url" content="https://SciML.github.io/NeuralLyapunov.jl/man/"/><meta property="twitter:url" content="https://SciML.github.io/NeuralLyapunov.jl/man/"/><link rel="canonical" href="https://SciML.github.io/NeuralLyapunov.jl/man/"/><script data-outdated-warner 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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="../">NeuralLyapunov.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><span class="tocitem">Manual</span><ul><li class="is-active"><a class="tocitem" href>Components of a Neural Lyapunov Problem</a></li><li><a class="tocitem" href="pdesystem/">Solving a Neural Lyapunov Problem</a></li><li><a class="tocitem" href="minimization/">Lyapunov Minimization Condition</a></li><li><a class="tocitem" href="decrease/">Lyapunov Decrease Condition</a></li><li><a class="tocitem" href="structure/">Structuring a Neural Lyapunov function</a></li><li><a class="tocitem" href="roa/">Training for Region of Attraction Identification</a></li><li><a class="tocitem" href="policy_search/">Policy Search and Network-Sependent Dynamics</a></li><li><a class="tocitem" href="local_lyapunov/">Local Lyapunov analysis</a></li></ul></li><li><span class="tocitem">Demonstrations</span><ul><li><a class="tocitem" href="../demos/damped_SHO/">Damped Simple Harmonic Oscillator</a></li><li><a class="tocitem" href="../demos/roa_estimation/">Estimating the Region of Attraction</a></li><li><a class="tocitem" href="../demos/policy_search/">Policy Search on the Driven Inverted Pendulum</a></li></ul></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><a class="is-disabled">Manual</a></li><li class="is-active"><a href>Components of a Neural Lyapunov Problem</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Components of a Neural Lyapunov Problem</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/SciML/NeuralLyapunov.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/SciML/NeuralLyapunov.jl/blob/master/docs/src/man.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="Components-of-a-Neural-Lyapunov-Problem"><a class="docs-heading-anchor" href="#Components-of-a-Neural-Lyapunov-Problem">Components of a Neural Lyapunov Problem</a><a id="Components-of-a-Neural-Lyapunov-Problem-1"></a><a class="docs-heading-anchor-permalink" href="#Components-of-a-Neural-Lyapunov-Problem" title="Permalink"></a></h1><p>For a candidate Lyapunov function <span>$V(x)$</span> to certify the stability of an equilibrium point <span>$x_0$</span> of the dynamical system <span>$\frac{dx}{dt} = f(x(t))$</span>, it must satisfy two conditions:</p><ol><li>The function <span>$V$</span> must be uniquely minimized at <span>$x_0$</span>, and </li><li>The function <span>$V$</span> must decrease along system trajectories (i.e., <span>$V(x(t))$</span> decreases as long as <span>$x(t)$</span> is a trajectory of the dynamical system).</li></ol><p>A neural Lyapunov function represents the candidate Lyapunov function <span>$V$</span> using a neural network, sometimes modifying the output of the network slightly so as to enforce one of the above conditions.</p><p>Thus, we specify our neural Lyapunov problems with three components, each answering a different question:</p><ol><li>How is <span>$V$</span> structured in terms of the neural network?</li><li>How is the minimization condition to be enforced?</li><li>How is the decrease condition to be enforced?</li></ol><p>These three components are represented by the three fields of a <a href="#NeuralLyapunov.NeuralLyapunovSpecification"><code>NeuralLyapunovSpecification</code></a> object.</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="NeuralLyapunov.NeuralLyapunovSpecification" href="#NeuralLyapunov.NeuralLyapunovSpecification"><code>NeuralLyapunov.NeuralLyapunovSpecification</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">NeuralLyapunovSpecification(structure, minimzation_condition, decrease_condition)</code></pre><p>Specifies a neural Lyapunov problem.</p><p><strong>Fields</strong></p><ul><li><code>structure</code>: a <a href="structure/#NeuralLyapunov.NeuralLyapunovStructure"><code>NeuralLyapunovStructure</code></a> specifying the relationship between the neural network and the candidate Lyapunov function.</li><li><code>minimzation_condition</code>: an <a href="minimization/#NeuralLyapunov.AbstractLyapunovMinimizationCondition"><code>AbstractLyapunovMinimizationCondition</code></a> specifying how the minimization condition will be enforced.</li><li><code>decrease_condition</code>: an <a href="decrease/#NeuralLyapunov.AbstractLyapunovDecreaseCondition"><code>AbstractLyapunovDecreaseCondition</code></a> specifying how the decrease condition will be enforced.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/NeuralLyapunov.jl/blob/ab3f1428265c9f62d9989fd3b7e4969877a911c6/src/conditions_specification.jl#L48-L60">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../">« Home</a><a class="docs-footer-nextpage" href="pdesystem/">Solving a Neural Lyapunov 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 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Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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Neural Lyapunov Problem</a></li><li><a class="tocitem" href="../minimization/">Lyapunov Minimization Condition</a></li><li><a class="tocitem" href="../decrease/">Lyapunov Decrease Condition</a></li><li><a class="tocitem" href="../structure/">Structuring a Neural Lyapunov function</a></li><li><a class="tocitem" href="../roa/">Training for Region of Attraction Identification</a></li><li><a class="tocitem" href="../policy_search/">Policy Search and Network-Sependent Dynamics</a></li><li><a class="tocitem" href="../local_lyapunov/">Local Lyapunov analysis</a></li><li class="is-active"><a class="tocitem" href>Internals</a></li></ul></li><li><span class="tocitem">Demonstrations</span><ul><li><a class="tocitem" href="../../demos/damped_SHO/">Damped Simple Harmonic Oscillator</a></li><li><a class="tocitem" href="../../demos/roa_estimation/">Estimating the Region of Attraction</a></li><li><a class="tocitem" href="../../demos/policy_search/">Policy Search on the Driven Inverted Pendulum</a></li></ul></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><a class="is-disabled">Manual</a></li><li class="is-active"><a href>Internals</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Internals</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/SciML/NeuralLyapunov.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/SciML/NeuralLyapunov.jl/blob/master/docs/src/man/internals.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="Internals"><a class="docs-heading-anchor" href="#Internals">Internals</a><a id="Internals-1"></a><a class="docs-heading-anchor-permalink" href="#Internals" title="Permalink"></a></h1><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="NeuralLyapunov.phi_to_net" href="#NeuralLyapunov.phi_to_net"><code>NeuralLyapunov.phi_to_net</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">phi_to_net(phi, θ[; idx])</code></pre><p>Return the network as a function of state alone.</p><p><strong>Arguments</strong></p><ul><li><code>phi</code>: the neural network, represented as <code>phi(x, θ)</code> if the neural network has a single output, or a <code>Vector</code> of the same with one entry per neural network output.</li><li><code>θ</code>: the parameters of the neural network; <code>θ[:φ1]</code> should be the parameters of the first neural network output (even if there is only one), <code>θ[:φ2]</code> the parameters of the second (if there are multiple), and so on.</li><li><code>idx</code>: the neural network outputs to include in the returned function; defaults to all and only applicable when <code>phi isa Vector</code>.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/NeuralLyapunov.jl/blob/ab3f1428265c9f62d9989fd3b7e4969877a911c6/src/numerical_lyapunov_functions.jl#L93-L106">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../local_lyapunov/">« Local Lyapunov analysis</a><a class="docs-footer-nextpage" href="../../demos/damped_SHO/">Damped Simple Harmonic Oscillator »</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 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