Algorithmic construction of Lyapunov functions for continuous vector fields via convex semi-infinite programs

TL;DR

This paper introduces a new computational method for constructing Lyapunov functions for nonlinear vector fields using convex semi-infinite programming, applicable without requiring explicit algebraic structure or analytical expressions.

Contribution

It proposes a numerically feasible algorithmic approach for Lyapunov function synthesis applicable to general nonlinear vector fields without special algebraic forms.

Findings

Successfully synthesizes Lyapunov functions for various nonlinear systems

Demonstrates the method's applicability without needing explicit system expressions

Provides numerical examples validating the approach

Abstract

This article presents a novel numerically tractable technique for synthesizing Lyapunov functions for equilibria of nonlinear vector fields. In broad strokes, corresponding to an isolated equilibrium point of a given vector field, a selection is made of a compact neighborhood of the equilibrium and a dictionary of functions in which a Lyapunov function is expected to lie. Then an algorithmic procedure based on the recent work [DACC22] is deployed on the preceding neighborhood-dictionary pair and charged with the task of finding a function satisfying a compact family of inequalities that defines the behavior of a Lyapunov function on the selected neighborhood. The technique applies to continuous nonlinear vector fields without special algebraic structures and does not even require their analytical expressions to proceed. Several numerical examples are presented to illustrate our results.