This paper studies the stability of discrete-time time-varying stochastic systems with infinite Markov switching. First, the concepts of strongly exponentially stable in mean square and exponentially stable in mean square with conditions are introduced, and their equivalent conditions are given by operator theory and stochastic analysis. Second, we introduce the Lyapunov equation related to strongly exponentially stable in mean square. Third, as an application of the proposed Lyapunov stability criterion, the relationship between the internal stability and the input–output state stability is derived for the infinite Markov jump system disturbed by finite-energy random disturbance. Finally, an example is given to verify its correctness and validity.
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