Can We Prove Stability by Using a Positive Definite Function with Non Sign-Definite Derivative?

TL;DR

This paper introduces new criteria for establishing the global asymptotic stability of nonlinear uncertain systems using positive definite functions that do not necessarily have negative semi-definite derivatives, expanding traditional Lyapunov methods.

Contribution

It presents a novel approach combining discretization and Matrosov's ideas to prove stability with positive definite functions lacking sign-definite derivatives.

Findings

New stability criteria for nonlinear systems

Applicable to functions without negative semi-definite derivatives

Illustrated with practical examples

Abstract

Novel criteria for global asymptotic stability of nonlinear uncertain finite-dimensional systems are presented. The results are obtained by a combination of the "discretization approach" and the ideas contained in the proof of the original Matrosov's result. The results can be used for the proof of global asymptotic stability by using continuously differentiable, positive definite functions which do not have a negative semi-definite derivative. Illustrating examples are provided.