Stability Analysis of Nonlinear Time-Varying Systems by Lyapunov Functions with Indefinite Derivatives

TL;DR

This paper extends Lyapunov stability analysis for nonlinear time-varying systems by allowing indefinite derivatives of Lyapunov functions, introducing scalar stable functions, and covering various stability types with numerical validation.

Contribution

It generalizes classical Lyapunov theorems to include indefinite derivatives and introduces scalar stable functions for broader stability analysis.

Findings

Generalized Lyapunov theorems for indefinite derivatives

Established stability criteria for various stability types

Validated results with numerical examples

Abstract

This paper is concerned with stability analysis of nonlinear time-varying systems by using Lyapunov function based approach. The classical Lyapunov stability theorems are generalized in the sense that the time-derivative of the Lyapunov functions are allowed to be indefinite. The stability analysis is accomplished with the help of the scalar stable functions introduced in our previous study. Both asymptotic stability and input-to-state stability are considered. Particularly, for asymptotic stability, several concepts such as uniform and non-uniform asymptotic stability, and uniform and non-uniform exponential stability are studied. The effectiveness of the proposed theorems is illustrated by several numerical examples.