Stability analysis of discrete-time stochastic systems with infinite Markov jump parameter

This paper investigates the stochastic stability and ℓ₂ input-state stability for a class of discrete-time infinite Markov jump systems with multiplicative noises. Based on a group of countably infinite coupled generalized Lyapunov equations/inequalities (ICGLEs/ICGLIs), a Lyapunov stability theorem is firstly presented for stochastic stability. Further, by means of detectability, an extended Lyapunov criterion is established, which relies on the positive semi-definite solution to a group of ICGLEs driven by a positive semi-definite term. Moreover, in the presence of finite-energy random disturbance, the relationship between the internal stability and ℓ₂ input-state stability of the considered systems is clarified.