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In mathematics, Lyapunov functions are functions which can be used to prove the stability of a certain fixed point in a dynamical system or autonomous differential equation. Named after the Russian mathematician Aleksandr Mikhailovich Lyapunov, Lyapunov functions are important to stability theory and control theory.## Definition of a Lyapunov candidate function

Let

$V$ is a Lyapunov-candidate-function if it is a locally positive-definite function, i.e.## Definition of the equilibrium point of a system

## Basic Lyapunov theorems for autonomous systems

### Stable equilibrium

If the Lyapunov-candidate-function $V$ is locally positive definite and the time derivative of the Lyapunov-candidate-function is locally negative semidefinite:
### Locally asymptotically stable equilibrium

If the Lyapunov-candidate-function $V$ is locally positive definite and the time derivative of the Lyapunov-candidate-function is locally negative definite:
### Globally asymptotically stable equilibrium

If the Lyapunov-candidate-function $V$ is globally positive definite, radially unbounded and the time derivative of the Lyapunov-candidate-function is globally negative definite:
## See also

## References

## External links

Functions which might prove the stability of some equilibrium are called Lyapunov-candidate-functions. There is no general method to construct or find a Lyapunov-candidate-function which proves the stability of an equilibrium, and the inability to find a Lyapunov function is inconclusive with respect to stability, which means, that not finding a Lyapunov function doesn't mean that the system is unstable. For dynamical systems (e.g. physical systems), conservation laws can often be used to construct a Lyapunov-candidate-function.

The basic Lyapunov theorems for autonomous systems which are directly related to Lyapunov (candidate) functions are a useful tool to prove the stability of an equilibrium of an autonomous dynamical system.

One must be aware that the basic Lyapunov Theorems for autonomous systems are a sufficient, but not necessary tool to prove the stability of an equilibrium. Finding a Lyapunov Function for a certain equilibrium might be a matter of luck. Trial and error is the method to apply, when testing Lyapunov-candidate-functions on some equilibrium.

- $V:mathbb\{R\}^n\; to\; mathbb\{R\}$

$V$ is a Lyapunov-candidate-function if it is a locally positive-definite function, i.e.

- $V(0)\; =\; 0\; ,$

- $V(x)\; >\; 0\; quad\; forall\; x\; in\; Usetminus\{0\}$

With $U$ being a neighborhood region around $x\; =\; 0$

Let

- $g\; :\; mathbb\{R\}^n\; to\; mathbb\{R\}^n$

- $dot\{y\}\; =\; g(y)\; ,$

- $0\; =\; g(y^*)\; ,$

There always exists a coordinate transformation $x\; =\; y\; -\; y^*\; ,$, such that:

- $dot\{x\}\; =\; g(x\; +\; y^*)\; =\; f(x)\; ,$

- $0\; =\; f(x^*)\; quad\; Rightarrow\; quad\; x^*\; =\; 0\; ,$

So the new system $f(x)$ has an equilibrium point at the origin.

Let

- $x^*\; =\; 0\; ,$

- $dot\{x\}\; =\; f(x)\; ,$

And let

- $dot\{V\}(x)\; =\; frac\{partial\; V\}\{partial\; x\}\; frac\{dx\}\{dt\}\; =\; nabla\; V\; dot\{x\}\; =\; nabla\; V\; f(x)$

- $dot\{V\}(x)\; le\; 0\; quad\; forall\; x\; in\; mathcal\{B\}$

- $dot\{V\}(x)\; <\; 0\; quad\; forall\; x\; in\; mathcal\{B\}setminus\{0\}$

- $dot\{V\}(x)\; <\; 0\; quad\; forall\; x\; in\; mathbb\{R\}^nsetminus\{0\},$

The Lyapunov-candidate function $V(x)$ is radially unbounded if

- $|\; x\; |\; to\; infty\; Rightarrow\; V(x)\; to\; infty$.

- Khalil, H.K. (1996).
*Nonlinear systems*. Prentice Hall Upper Saddle River, NJ.

- Example of determining the stability of the equilibrium solution of a system of ODEs with a Lyapunov function

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This article is licensed under the GNU Free Documentation License.

Last updated on Friday October 03, 2008 at 10:27:23 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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