Stability and bounded real lemmas of discrete-time MJLSs with the Markov chain on a Borel space

In this paper, exponential stability of discrete-time Markov jump linear systems (MJLSs) with the Markov chain on a Borel space $(\Theta ,B\left(\Theta \right))$ is studied, and bounded real lemmas (BRLs) are given. The work generalizes the results from the previous literature that considered only the Markov chain taking values in a countable set to the scenario of an uncountable set and provides unified approaches for describing exponential stability and $H_{\infty }$ performance of MJLSs. This paper covers two kinds of exponential stabilities: one is exponential mean-square stability with conditioning (EMSSy-C), and the other is exponential mean-square stability (EMSSy). First, based on the infinite-dimensional operator theory, the equivalent conditions for determining these two kinds of stabilities are shown respectively by the exponentially stable evolutions generated by the corresponding bounded linear operators on different Banach spaces, which turn out to present the spectral criteria of EMSSy-C and EMSSy. Furthermore, the relationship between these two kinds of stabilities is discussed. Moreover, some easier-to-check criteria are established for EMSSy-C of MJLSs in terms of the existence of uniformly positive definite solutions of Lyapunov-type equations or inequalities. In addition, BRLs are given separately in terms of the existence of solutions of the $\Theta$-coupled difference Riccati equation for the finite horizon case and algebraic Riccati equation for the infinite horizon case, which facilitates the $H_{\infty }$ analysis of MJLSs with the Markov chain on a Borel space.