Questions tagged [lyapunov-functions]

Named after the Russian mathematician Aleksandr Mikhailovich Lyapunov, Lyapunov functions (also called the Lyapunov’s second method for stability) are important to stability theory of dynamical systems and control theory. A similar concept appears in the theory of general state space Markov chains, usually under the name Foster–Lyapunov functions.

A Lyapunov function for an autonomous dynamical system $$ \begin{cases} g:\mathbb{R}^n \to \mathbb{R}^n \\ y'=g(y)\end{cases}$$ with an equilibrium point at ${\displaystyle y=0}$ is a scalar function ${\displaystyle V:\mathbb {R} ^{n}\to \mathbb {R} }$ that is continuous, has continuous first derivatives, is locally positive-definite, and for which ${\displaystyle -\nabla {V}\cdot g}$ is also locally positive definite. The condition that ${\displaystyle -\nabla {V}\cdot g}$ is locally positive definite is sometimes stated as ${\displaystyle \nabla {V}\cdot g}$ is locally negative definite.

Reference:

https://en.wikipedia.org/wiki/Lyapunov_function