Lyapunov function

A Lyapunov function is a scalar function defined on the state space of a dynamical system that enables the analysis of stability without solving the system's equations of motion. Named after Aleksandr Lyapunov, it generalizes the intuitive notion of energy: it is positive everywhere except at an equilibrium point, and its rate of change along system trajectories is negative. The existence of such a function guarantees Lyapunov stability or asymptotic stability; its non-existence tells us only that the energy-landscape method does not apply, not that the system is unstable.

The construction of Lyapunov functions for nonlinear systems remains an art rather than an algorithm. For linear systems, quadratic forms suffice; for mechanical systems, total energy often works; for general nonlinear systems, one may need to search through classes of candidate functions using sum-of-squares optimization or machine learning approaches. A control-Lyapunov function is a Lyapunov function for which an explicit stabilizing control law can be derived, forming the bridge between stability analysis and controller design.