# Lyapunov function

### From Exampleproblems

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.

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## Definition

Given an autonomous system of two first order differential equations:

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

A function *V*(*x*,*y*) that is of class *C*^{1} and satisfies *V*(0,0) = 0 is called a **Lyapunov function** if every open ball *B*_{δ}(0,0) contains at least one point where *V* > 0. If there happens to exist δ^{ * } such that the function , given by

is positive definite in , then the origin is an unstable critical point of the system.

## See also

## References

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

## External Links

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