Asymptotic stability implies the existence of a local polynomial Lyapunov function

TL;DR

This paper proves that globally asymptotically stable systems with smooth vector fields always have a local polynomial Lyapunov function near the equilibrium point, aiding stability analysis.

Contribution

It establishes the existence of local polynomial Lyapunov functions for a broad class of stable systems, bridging a gap in stability theory.

Findings

Existence of local polynomial Lyapunov functions for stable systems

Applicability to systems with twice continuously differentiable vector fields

Supports stability verification through polynomial Lyapunov functions

Abstract

We show that every globally asymptotically stable system with a twice continuously differentiable vector field admits a local polynomial Lyapunov function on an arbitrary bounded neighborhood of the origin.