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

This technical note proposes a unified Lyapunov framework for analyzing the stochastic asymptotic and finite-time convergence/stability for Ito 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 non-zero 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.