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.
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 , given by
- Example of determining the stability of the equilibrium solution of a system of ODEs with a Lyapunov function