Equivalence of Finite- and Fixed-time Stability to Asymptotic Stability

TL;DR

This paper introduces new Lyapunov-like methods for analyzing finite- and fixed-time stability in dynamical systems, demonstrating how asymptotic stability can be transformed into fixed-time stability through scaling.

Contribution

It provides the first and second-order non-smooth Lyapunov results for finite- and fixed-time convergence, relaxing the need for differentiable Lyapunov functions.

Findings

Finite- and fixed-time convergence can be characterized using non-smooth Lyapunov functions.

A globally asymptotically stable system can be scaled to achieve fixed-time stability.

Convergence rates are interconnected via suitable transformations.

Abstract

In this paper, we present new results on finite- and fixed-time convergence for dynamical systems using LaSalle-like invariance principles. In particular, we provide first and second-order non-smooth Lyapunov-like results for finite- and fixed-time convergence, thereby relaxing the requirement of existence a differentiable, positive definite Lyapunov function. Based on these findings, we show that a dynamical system whose equilibrium point is globally asymptotically stable can be modified through scaling so that the resulting dynamical system has a fixed-time stable equilibrium point. The results in this paper expand our understanding of various convergence rates and strengthen the hypothesis that all the convergence rates are interconnected through a suitable transformation.