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.

1 Definition
2 See also
3 References
4 External Links

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 $C 1$ 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 $\dot{V}$ , given by