Study of Stability of Discrete-Time Stochastic Systems with Time Variations Through Lyapunov's Method

Abstract

In this paper, theoretical investigations have been carried out on the stability analysis of discrete time-varying stochastic equations, using the Lyapunov direct second method. Starting with the discrete time-varying stochastic system derived from the Ito-integrated formula, we apply a quadratic Lyapunov function and utilize Lyapunov theorems to assess the stochastic stability of the trivial (zero) solution. In particular, we analyze its $p$-stability, mean-square stability, and stochastic asymptotic stability as time approaches infinity. Lastly, we demonstrate these methods through comprehensive examples.