# Lyapunov function

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In the theory of dynamical systems, and control theory, Lyapunov functions, named after Aleksandr Mikhailovich Lyapunov, are a family of functions that can be used to demonstrate the stability or instability of some state points of a system.

The demonstration of stability or instability requires finding a Lyapunov function for that system. There is no direct method to obtain a Lyapunov function but there may be tricks to simplify the task. The inability to find a Lyapunov function is inconclusive with respect to stability or instability.

## Definition

Given an autonomous system of two first order differential equations:

$\frac{dx}{dt}=F(x,y) \quad \frac{dy}{dt}=G(x,y)$

Let the origin (0,0) be an isolated critical point of the above system.

A function V(x,y) that is of class C1 and satisfies V(0,0) = 0 is called a Lyapunov function if every open ball Bδ(0,0) contains at least one point where V > 0. If there happens to exist δ * such that the function $\dot{V}$, given by

$\dot{V}(x,y)=V_{x}(x,y)F(x,y)+V_{y}(x,y)G(x,y)$

is positive definite in $B_{\delta^*}(0,0)$, then the origin is an unstable critical point of the system.

## References

This article incorporates material from Liapunov function on PlanetMath, which is licensed under the GFDL.Template:Mathapplied-stub

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