Stochastic Stability of Jump Discrete-Time Linear Systems With Markov Chain in a General Borel Space

Necessary and sufficient conditions for stochastic stability (SS) of discrete-time linear systems subject to Markov jumps in the parameters are considered, assuming that the Markov chain takes values in a general Borel space S. It is shown that SS is equivalent to the spectrum radius of a bounded linear operator in a Banach space being less than 1, or to the existence of a solution of a Lyapunov type equation. These results generalize several previous results in the literature, which considered only the case of the Markov chain taking values in a finite or infinite countable space.