李雅普诺夫函数（Lyapunov function）是用来证明一动力系统或自治微分方程稳定性的函数。其名称来自俄罗斯数学家亚历山大·李亚普诺夫（Aleksandr Mikhailovich Lyapunov）。李亚普诺夫函数在稳定性理论及控制理论中相当重要。

若一函数可能可以证明系统在某平衡点的稳定性，此函数称为李亚普诺夫候选函数（Lyapunov-candidate-function）。不过目前还找不到一般性的方式可建构（或找到）一个系统的李亚普诺夫候选函数，而找不到李亚普诺夫函数也不代表此系统不稳定。在动态系统中，有时会利用守恒律来建构李亚普诺夫候选函数。

In the theory of ordinary differential equations (ODEs), Lyapunov functions are scalar functions that may be used to prove the stability of an equilibrium of an ODE. Named after the Russian mathematician Aleksandr Mikhailovich Lyapunov, who introduced them in his doctoral thesis General Problem of the Stability of Motion, the method of Lyapunov functions (also called the Lyapunov’s second method for stability) is important to stability theory of dynamical systems and control theory. Actually, it is the only universal method for the investigation of the stability of nonlinear dynamical systems of general configuration.