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

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.