Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Lyapunov function
open-in-new
<h2 id="definition">Definition</h2> <p>A Lyapunov function for an autonomous <a href="/facts/Dynamical_system/782gREl5">dynamical system</a> </p> { g : R n → R n y ˙ = g ( y ) {\displaystyle {\begin{cases}g:\mathbb {R} ^{n}\to \mathbb {R} ^{n}&\\{\dot {y}}=g(y)\end{cases}}} <p>with an equilibrium point at y = 0 {\displaystyle y=0} is a <a href="/facts/Scalar_function/8MtPTGob">scalar function</a> V : R n → R {\displaystyle V:\mathbb {R} ^{n}\to \mathbb {R} } that is continuous, has continuous first derivatives, is strictly positive for y ≠ 0 {\displaystyle y\neq 0} , and for which the time derivative V ˙ = ∇ V ⋅ g {\displaystyle {\dot {V}}=\nabla {V}\cdot g} is non positive (these conditions are required on some region containing the origin). The (stronger) condition that − ∇ V ⋅ g {\displaystyle -\nabla {V}\cdot g} is strictly positive for y ≠ 0 {\displaystyle y\neq 0} is sometimes stated as − ∇ V ⋅ g {\displaystyle -\nabla {V}\cdot g} is <i>locally positive definite</i>, or ∇ V ⋅ g {\displaystyle \nabla {V}\cdot g} is <i>locally negative definite</i>. </p> <h3>Further discussion of the terms arising in the definition</h3> <p>Lyapunov functions arise in the study of equilibrium points of dynamical systems. In R n , {\displaystyle \mathbb {R} ^{n},} an arbitrary autonomous <a href="/facts/Dynamical_system/782gREl5">dynamical system</a> can be written as </p> y ˙ = g ( y ) {\displaystyle {\dot {y}}=g(y)} <p>for some smooth g : R n → R n . {\displaystyle g:\mathbb {R} ^{n}\to \mathbb {R} ^{n}.} </p><p>An equilibrium point is a point y ∗ {\displaystyle y^{*}} such that g ( y ∗ ) = 0. {\displaystyle g\left(y^{*}\right)=0.} Given an equilibrium point, y ∗ , {\displaystyle y^{*},} there always exists a coordinate transformation x = y − y ∗ , {\displaystyle x=y-y^{*},} such that: </p> { x ˙ = y ˙ = g ( y ) = g ( x + y ∗ ) = f ( x ) f ( 0 ) = 0 {\displaystyle {\begin{cases}{\dot {x}}={\dot {y}}=g(y)=g\left(x+y^{*}\right)=f(x)\\f(0)=0\end{cases}}} <p>Thus, in studying equilibrium points, it is sufficient to assume the equilibrium point occurs at 0 {\displaystyle 0} . </p><p>By the chain rule, for any function, H : R n → R , {\displaystyle H:\mathbb {R} ^{n}\to \mathbb {R} ,} the time derivative of the function evaluated along a solution of the dynamical system is </p> H ˙ = d d t H ( x ( t ) ) = ∂ H ∂ x ⋅ d x d t = ∇ H ⋅ x ˙ = ∇ H ⋅ f ( x ) . {\displaystyle {\dot {H}}={\frac {d}{dt}}H(x(t))={\frac {\partial H}{\partial x}}\cdot {\frac {dx}{dt}}=\nabla H\cdot {\dot {x}}=\nabla H\cdot f(x).} <p>A function H {\displaystyle H} is defined to be locally <a href="/facts/Positive-definite_function/RKqaCfp9">positive-definite function</a> (in the sense of dynamical systems) if both H ( 0 ) = 0 {\displaystyle H(0)=0} and there is a neighborhood of the origin, B {\displaystyle {\mathcal {B}}} , such that: </p> H ( x ) > 0 ∀ x ∈ B ∖ { 0 } . {\displaystyle H(x)>0\quad \forall x\in {\mathcal {B}}\setminus \{0\}.} <h2 id="basic-lyapunov-theorems-for-autonomous-systems">Basic Lyapunov theorems for autonomous systems</h2> <p class="note">Main article: <a href="/facts/Lyapunov_stability/r3je5zaQ">Lyapunov stability</a></p> <p>Let x ∗ = 0 {\displaystyle x^{*}=0} be an equilibrium point of the autonomous system </p> x ˙ = f ( x ) . {\displaystyle {\dot {x}}=f(x).} <p>and use the notation V ˙ ( x ) {\displaystyle {\dot {V}}(x)} to denote the time derivative of the Lyapunov-candidate-function V {\displaystyle V} : </p> V ˙ ( x ) = d d t V ( x ( t ) ) = ∂ V ∂ x ⋅ d x d t = ∇ V ⋅ x ˙ = ∇ V ⋅ f ( x ) . {\displaystyle {\dot {V}}(x)={\frac {d}{dt}}V(x(t))={\frac {\partial V}{\partial x}}\cdot {\frac {dx}{dt}}=\nabla V\cdot {\dot {x}}=\nabla V\cdot f(x).} <h3>Locally asymptotically stable equilibrium</h3> <p>If the equilibrium point is isolated, the Lyapunov-candidate-function V {\displaystyle V} is locally positive definite, and the time derivative of the Lyapunov-candidate-function is locally negative definite: </p> V ˙ ( x ) < 0 ∀ x ∈ B ( 0 ) ∖ { 0 } , {\displaystyle {\dot {V}}(x)<0\quad \forall x\in {\mathcal {B}}(0)\setminus \{0\},} <p>for some neighborhood B ( 0 ) {\displaystyle {\mathcal {B}}(0)} of origin, then the equilibrium is proven to be locally asymptotically stable. </p> <h3>Stable equilibrium</h3> <p>If V {\displaystyle V} is a Lyapunov function, then the equilibrium is <a href="/facts/Lyapunov_stable/r3je5zaQ">Lyapunov stable</a>. </p> <h3>Globally asymptotically stable equilibrium</h3> <p>If the Lyapunov-candidate-function V {\displaystyle V} is globally positive definite, <a href="/facts/Radially_unbounded_function/HmMXo1cT">radially unbounded</a>, the equilibrium isolated and the time derivative of the Lyapunov-candidate-function is globally negative definite: </p> V ˙ ( x ) < 0 ∀ x ∈ R n ∖ { 0 } , {\displaystyle {\dot {V}}(x)<0\quad \forall x\in \mathbb {R} ^{n}\setminus \{0\},} <p>then the equilibrium is proven to be <a href="/facts/Stability_theory/Su5sHnxs">globally asymptotically stable</a>. </p><p>The Lyapunov-candidate function V ( x ) {\displaystyle V(x)} is radially unbounded if </p> ‖ x ‖ → ∞ ⇒ V ( x ) → ∞ . {\displaystyle \|x\|\to \infty \Rightarrow V(x)\to \infty .} <p>(This is also referred to as norm-coercivity.) </p><p>The converse is also true,<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> and was proved by <a href="/facts/Jos%25C3%25A9_Luis_Massera/fTd5Joaa">José Luis Massera</a> (see also <a href="/facts/Massera%2527s_lemma/mNekXEUL">Massera's lemma</a>). </p> <h2 id="example">Example</h2> <p>Consider the following differential equation on R {\displaystyle \mathbb {R} } : </p> x ˙ = − x . {\displaystyle {\dot {x}}=-x.} <p>Considering that x 2 {\displaystyle x^{2}} is always positive around the origin it is a natural candidate to be a Lyapunov function to help us study x {\displaystyle x} . So let V ( x ) = x 2 {\displaystyle V(x)=x^{2}} on R {\displaystyle \mathbb {R} } . Then, </p> V ˙ ( x ) = V ′ ( x ) x ˙ = 2 x ⋅ ( − x ) = − 2 x 2 < 0. {\displaystyle {\dot {V}}(x)=V'(x){\dot {x}}=2x\cdot (-x)=-2x^{2}<0.} <p>This correctly shows that the above differential equation, x , {\displaystyle x,} is asymptotically stable about the origin. Note that using the same Lyapunov candidate one can show that the equilibrium is also globally asymptotically stable. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Lyapunov_stability/r3je5zaQ">Lyapunov stability</a></li> <li><a href="/facts/Ordinary_differential_equation/ygKGQ8kD">Ordinary differential equations</a></li> <li><a href="/facts/Control-Lyapunov_function/scjr1L5Y">Control-Lyapunov function</a></li> <li><a href="/facts/Chetaev_function/4w5wP1ek">Chetaev function</a></li> <li><a href="/facts/Foster%2527s_theorem/UNK4tJAz">Foster's theorem</a></li> <li><a href="/facts/Lyapunov_optimization/zAXmbrsI">Lyapunov optimization</a></li></ul> <ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/LyapunovFunction.html">"Lyapunov Function"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li> <li>Khalil, H.K. (1996). <i>Nonlinear systems</i>. Prentice Hall Upper Saddle River, NJ.</li> <li>La Salle, Joseph; Lefschetz, Solomon (1961). <i>Stability by Liapunov's Direct Method: With Applications</i>. New York: Academic Press.</li> <li><i>This article incorporates material from Lyapunov function on <a href="/facts/PlanetMath/wphPCtLf">PlanetMath</a>, which is licensed under the Creative Commons Attribution/Share-Alike License.</i></li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="https://web.archive.org/web/20110926230621/http://www.exampleproblems.com/wiki/index.php/ODELF1">Example</a> of determining the stability of the equilibrium solution of a system of ODEs with a Lyapunov function</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Massera, José Luis (1949), "On Liapounoff's conditions of stability", Annals of Mathematics, Second Series, 50 (3): 705–721, doi:10.2307/1969558, JSTOR 1969558, MR 0035354 <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> </ol>