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In the theory of [[ordinary differential equations]] (ODEs), '''Lyapunov functions''' are scalar functions that may be used to prove the stability of an [[equilibrium point|equilibrium]] of an ODE. Named after the [[Russia]]n [[mathematician]] [[Aleksandr Lyapunov|Aleksandr Mikhailovich Lyapunov]], Lyapunov functions are important to [[stability theory]] and [[control theory]]. A similar concept appears in the theory of general state space [[Markov Chain]]s, usually under the name [[Foster-Lyapunov functions]].
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For many classes of ODEs, the existence of Lyapunov functions is a necessary and sufficient condition for stability. Whereas there is no general technique for constructing Lyapunov functions for ODEs, in many specific cases, the construction of Lyapunov functions is known. For instance, [[quadratic function|quadratic]] functions suffice for systems with one state; the solution of a particular [[linear matrix inequality]] provides Lyapunov functions for linear systems; and [[conservation law]]s can often be used to construct Lyapunov functions for [[physical system]]s.
 
Informally, a Lyapunov function is a function that takes positive values everywhere except at the equilibrium in question, and decreases (or is non-increasing) along ''every'' trajectory of the ODE. The principal merit of Lyapunov function-based stability analysis of ODEs is that the actual solution (whether analytical or numerical) of the ODE is not required.
 
== Definition of a Lyapunov candidate function==
Let
:<math>V:\mathbb{R}^n \to \mathbb{R}</math>
be a continuous [[scalar function]]. <br>
<math>V</math> is a Lyapunov-candidate-function if it is a locally [[positive-definite function]], i.e.
 
:<math>V(0) = 0 \,</math>
:<math>V(x) > 0 \quad \forall x \in U\setminus\{0\}</math>
 
with <math>U</math> being a neighborhood region around <math>x = 0.</math>
 
== Definition of the equilibrium point of a system==
Let
:<math>g : \mathbb{R}^n \to \mathbb{R}^n</math>
:<math>\dot{y} = g(y) \,</math>
be an arbitrary autonomous [[dynamical system]] with equilibrium point <math>y^* \,</math>:
:<math>0 = g(y^*). \,</math>
 
There always exists a coordinate transformation <math>x = y - y^* \,</math>, such that:
:<math>\dot{x} = g(x + y^*) = f(x) \,</math>
:<math> f(0) = 0. \,</math>
 
So the new system <math>f(x)</math> has an equilibrium point at the origin.
 
== Basic Lyapunov theorems for autonomous systems==
:{{main|Lyapunov stability}}
 
Let
:<math>x^* = 0 \,</math>
be an equilibrium of the autonomous system
:<math>\dot{x} = f(x). \,</math>
 
And let
:<math>\dot{V}(x) = \frac{\partial V}{\partial x}\cdot \frac{dx}{dt} = \nabla V \cdot \dot{x} = \nabla V\cdot f(x) = \frac{d}{dt} V(x(t)) </math>
be the time derivative of the Lyapunov-candidate-function <math>V</math>.
 
===Stable equilibrium===
If the Lyapunov-candidate-function <math>V</math> is locally positive definite and the time derivative of the Lyapunov-candidate-function is locally negative semidefinite:
:<math>\dot{V}(x) \leq 0 \quad \forall x \in \mathcal{B}\setminus\{0\}</math>
for some neighborhood <math>\mathcal{B}</math> of <math>0</math>, then the equilibrium is proven to be [[Stability theory|stable]].
 
===Locally asymptotically stable equilibrium===
If the Lyapunov-candidate-function <math>V</math> is locally positive definite and the time derivative of the Lyapunov-candidate-function is locally negative definite:
:<math>\dot{V}(x) < 0 \quad \forall x \in \mathcal{B}\setminus\{0\}</math>
for some neighborhood <math>\mathcal{B}</math> of <math>0</math>, then the equilibrium is proven to be [[Stability theory|locally asymptotically stable]]. The converse is also true, and was proved by [[José Luis Massera|J. L. Massera]]
 
===Globally asymptotically stable equilibrium===
If the Lyapunov-candidate-function <math>V</math> is globally positive definite, [[Radially unbounded function|radially unbounded]] and the time derivative of the Lyapunov-candidate-function is globally negative definite:
:<math>\dot{V}(x) < 0 \quad \forall x \in \mathbb{R}^n\setminus\{0\},</math>
then the equilibrium is proven to be [[Stability theory|globally asymptotically stable]].
 
The Lyapunov-candidate function <math>V(x)</math> is radially unbounded if
:<math>\| x \| \to \infty  \Rightarrow V(x) \to \infty. </math>
(This is also referred to as norm-coercivity.)
 
==Example==
Consider the following differential equation with solution x on <math>\mathbb{R}</math>:
:<math>\dot x = -x.</math>
Considering that |x| is always positive around the origin it is a natural candidate to be a Lyapunov function to help us study x.
So let <math>V(x)=|x|</math> on <math>\mathbb{R}\setminus\{0\}</math>. Then,
:<math>\dot V(x) = V'(x) f(x) = \mathrm{sgn}(x)\cdot (-x) = -|x|<0.</math>
This correctly shows that the above differential equation, x, is asymptotically stable about the origin.
 
==See also==
* [[Ordinary differential equation]]s
* [[Control-Lyapunov function]]
* [[Foster's theorem]]
* [[Lyapunov optimization]]
 
==References==
* {{mathworld|urlname=LyapunovFunction|title= Lyapunov Function}}
* {{cite book
| author = Khalil, H.K.
| year = 1996
| title = Nonlinear systems
| publisher = Prentice Hall Upper Saddle River, NJ
| isbn =
}}
* {{PlanetMath attribution|id=4386|title=Lyapunov function}}
 
==External links==
* [http://www.exampleproblems.com/wiki/index.php/ODELF1 Example] of determining the stability of the equilibrium solution of a system of ODEs with a Lyapunov function
* [http://www.efg2.com/Lab/FractalsAndChaos/Lyapunov.htm Some Lyapunov diagrams]
 
[[Category:Stability theory]]

Revision as of 23:33, 27 February 2014

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