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

Kunal Garg

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.

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

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.
Paper Structure (6 sections, 11 theorems, 30 equations)

This paper contains 6 sections, 11 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Mathematical Preliminaries
  3. New results on FTS and FxTS
  4. Stability analysis for modified dynamics
  5. Conclusions
  6. Acknowledgments

Key Result

Lemma 1

cortes2006finite Let $x:\mathbb R_+\to \mathbb R^n$ be a Fillipov solution of eq: sys and let $V:\mathbb R^n\to \mathbb R$ be a locally Lipschitz and regular function. Then, $V\circ x:\mathbb R_+\to \mathbb R$ is absolutely continuous and $\frac{d}{dt}V(x(t))\in \tilde{L}_fV(\phi(t))$ almost everywh

Theorems & Definitions (21)

  • Lemma 1
  • Lemma 2
  • Definition 1
  • Lemma 3
  • Theorem 1
  • Lemma 4
  • Remark 1
  • Corollary 1
  • Corollary 2
  • Remark 2
  • ...and 11 more