Optimal control for infinite-dimensional linear systems with Markovian jumps in Borel spaces

This paper addresses an optimal control problem for discrete-time, infinite - dimensional linear systems with Markovian jumps (LSMJs) in a Borel space. The main objective is to minimize an infinite-horizon, non-negative quadratic cost functional for time-dependent LSMJs by generalizing previous findings on finite - dimensional systems to the infinite-dimensional case. A second objective is to establish a framework that extends known results concerning the existence of certain global solutions (such as minimal, maximal, and stabilizing solutions) for a more general class of Riccati equations associated with the optimal control problem. Key results include the convergence properties of the zero-final-value solutions of the associated backward generalized Riccati equations (GREs) and Lyapunov-type stability criteria for LSMJs. The optimal control law and the corresponding optimal cost are then computed using the minimal solution of the GRE.