This tag is for questions relating to Lyapunov function, which is a scalar function defined on the phase space, which can be used to prove the stability of an equilibrium point. The Lyapunov function method is applied to study the stability of various differential equations and systems.
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.
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