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This paper is concerned with a kind of discrete-time stochastic systems with Markov jump parameters taking values in a Borel measurable set. First, both strong exponential stability and exponential stability in the mean square sense are introduced for the considered systems. Based on generalized Lyapunov equation and inequality, necessary and sufficient conditions are derived for the strong exponential stability. By use of the given stability criteria, it is shown that strong exponential stability can lead to exponential stability and further to stochastic stability. Moreover, strong exponential stability can guarantee the so-called l 2 input-state stability, which characterizes the asymptotic behavior of system state influenced by exogenous disturbance with finite energy. Second, H ∞ performance is analyzed for the perturbed dynamic models over finite and infinite horizons, respectively. For a prescribed disturbance attenuation level, stochastic bound real lemmas are presented in terms of Riccati equations or linear matrix inequalities. As a direct application, the infinite-horizon H ∞ control problem is settled and the state-feedback controller is constructed. Numerical simulations are conducted to illustrate the validity of the proposed results. INDEX TERMS Markov chain, Borel set, stability, bounded real lemma, H ∞ control.

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