Study of stability of discrete-time stochastic systems with time variations through Lyapunov’s method

In this paper, we carry out a comprehensive stability analysis of discrete time-varying stochastic equations using the Lyapunov direct second method. By constructing an appropriate quadratic Lyapunov function, we successfully apply Lyapunov?s theorems to investigate the stochastic stability of the trivial solution of the system. Our analysis led to significant findings regarding the system?s p-stability, mean-square stability, and stochastic asymptotic stability in the large. The results of this research underscore the effectiveness and versatility of the Lyapunov direct second method in assessing the stability characteristics of complex stochastic systems. In particular, we establish that, under suitable conditions, the trivial solu-tion satisfies key stability properties, namely p-stability, mean-square stability, and stochastic asymptotic stability. These findings are not only theoretically significant, but also have substantial practical relevance. The methods developed in this work are lastly illustrated through comprehensive examples.