Improved finite-time stability and instability theorems for stochastic nonlinear systems

TL;DR

This paper introduces new theorems for finite-time stability and instability of stochastic nonlinear systems, relaxing previous constraints and broadening applicability, supported by simulations.

Contribution

It proposes improved criteria for finite-time stability that allow indefinite Lyapunov infinitesimal operators, extending existing results.

Findings

New sufficient condition for global solutions.

Relaxed stability criteria allowing indefinite $\\mathcal{L}V$.

Simulation examples verify theoretical results.

Abstract

This paper studies finite-time stability and instability theorems in probability sense for stochastic nonlinear systems. Firstly, a new sufficient condition is proposed to guarantee that the considered system has a global solution. Secondly, we propose improved finite-time stability and instability criteria that relax the constraints on $\mathcal {L}V$ (the infinitesimal operator of Lyapunov function $V$) by the uniformly asymptotically stable function(UASF). The improved finite-time stability theorems allow $\mathcal {L}V$ to be indefinite (negative or positive) rather than just only allow $\mathcal {L}V$ to be negative. Most existing finite-time stability and instability results can be viewed as special cases of the obtained theorems. Finally, some simulation examples verify the validity of the theoretical results.