A Note on Convergence in Maximal Solution Problems for Infinite Markov Jump Linear Systems

In the context of infinite Markov jump linear systems (IMJLS), stochastic stability (a sort of L2-stability) is a structural concept intimately related to a certain bounded linear operator (D). Infinite (or finite) here, has to do with the state space of the Markov chain being finite or infinite countable. In the path to solving the maximal solution problem in the infinite countable case, a certain sequence of bounded linear operators (which converges trivially to D in the finite case) arises and convergence in the norm topology (uniform operator topology) becomes a relevant point. In this paper, we provide a condition that insures that this convergence also holds in the infinite countable case. This condition is automaticaly satisfied when we reduce the problem to the finite case. The issue of whether this is a restrictive condition or not, is brought to light using arguments that stems from the probabilistic nature of the Markovchain. This, in conjunction with a class of counterexamples, unveil further differences between the finite and the infinite countable case. We also establish a (weaker) condition for the spectrum of the limit of the above sequence of operators being in the closed left half-plane of the complex numbers.