Optimal control for linear discrete-time systems with Markov perturbations in Hilbert spaces

In this article, we discuss a quadratic control problem for linear discrete-time systems with Markov perturbations in Hilbert spaces, which is linked to a discrete-time Riccati equation defined on certain infinite-dimensional ordered Banach space. We prove that under stabilizability and stochastic uniform observability conditions, the Riccati equation has a unique, uniformly positive, bounded on N and stabilizing solution. Based on this result, we solve the proposed optimal control problem. An example illustrates the theory.