Unified Stability Analysis for Itô Stochastic Systems: From Almost Surely Asymptotic to Finite-Time Convergence

This article proposes a unified Lyapunov framework for analyzing the stochastic asymptotic and finite-time convergence/stability for Itô stochastic nonlinear systems. By exploring the coupling effect between the drift and the diffusion parts of the system, novel almost sure convergence/stability criteria are established. For the finite-time case, the stability criteria not only capture the stabilizing effect of stochastic noise but also include the existing finite-time stability criteria as special cases. For the asymptotic case, it removes the local Lipschitz conditions and the nonzero property of the solution demanded by the existing results. The proposed theoretical results are further applied to solve the sliding mode control and the optimal finite-time/asymptotic stabilization problems.