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:
- <math> \frac{dx}{dt}=F(x,y)\quad\frac{dy}{dt}=G(x,y)</math>
Let the origin (0,0) be an isolated critical point of the above system.
A function <math> V(x,y)</math> that is of class <math>C^{1}</math> and satisfies <math>V(0,0)=0</math> is called a Lyapunov function if every open ball <math> B_\delta(0,0)</math> contains at least one point where <math> V>0</math>. If there happens to exist <math> \delta^*</math> such that the function <math> \dot{V}</math>, given by
- <math>\dot{V}(x,y)=V_{x}(x,y)F(x,y)+V_{y}(x,y)G(x,y) </math>
is positive definite in <math> B_{\delta^*}(0,0) </math>, then the origin is an unstable critical point of the system.
See also
External links
- Lyapunov Function from Wolfram Research's MathWorld