In this paper, exponential stability of discrete-time Markov jump linear systems (MJLSs) with the Markov chain on a Borel space 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 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 -coupled difference Riccati equation for the finite horizon case and algebraic Riccati equation for the infinite horizon case, which facilitates the analysis of MJLSs with the Markov chain on a Borel space.

中文翻译：

---

Borel 空间上马尔可夫链离散时间 MJLS 的稳定性和有界实引理


本文研究了Borel空间上带有马尔可夫链的离散时间马尔可夫跳跃线性系统（MJLS）的指数稳定性，并给出了有界实数引理（BRL）。这项工作将之前仅考虑马尔可夫链在可数集合中取值的文献的结果推广到不可数集合的场景，并提供了描述 MJLS 的指数稳定性和性能的统一方法。本文涵盖两种指数稳定性：一种是带条件的指数均方稳定性（EMSSy-C），另一种是指数均方稳定性（EMSSy）。首先，基于无限维算子理论，通过相应的有界线性算子在不同Banach空间上产生的指数稳定演化分别给出了确定这两种稳定性的等效条件，从而给出了谱判据EMSSy-C 和 EMSSy。此外，还讨论了这两种稳定性之间的关系。此外，根据Lyapunov型方程或不等式的一致正定解的存在性，为MJLS的EMSSy-C建立了一些更容易检查的准则。此外，根据有限层情况下的耦合差分Riccati方程和无限层情况下的代数Riccati方程解的存在性分别给出了BRL，这有利于Borel空间上马尔可夫链的MJLS的分析。