Stochastic Singular Linear Systems and Related Linear-Quadratic Optimal Control Problems under Finite and Infinite Horizons

Abstract: In this paper, we study the necessary and sufficient conditions for ensuring the well-posedness of the stochastic singular systems. Moreover, we investigate the stochastic singular linear-quadratic control problems, considering both finite and infinite horizons, and transform each of these problems into their corresponding normal linear-quadratic control problem. To guarantee the finiteness of the infinite-horizon linear-quadratic control problem, we establish the Popov-Belevitch-Hautus rank criterion for accessing the controllability of the stochastic system. Furthermore, we derive the feedback form of the optimal control. Finally, we provide solutions for illustrated examples of the stochastic singular linear-quadratic control problem in both finite and infinite horizons.