Indefinite Linear Quadratic Optimal Control and Stabilization Problem for Discrete-Time Rectangular Descriptor Markov Jump Systems With Noise

This work investigates the indefinite linear quadratic (LQ) optimal control problem for discrete-time rectangular descriptor Markov jump systems (DMJSs) with additive noise on finite and infinite horizon, where the weight matrices of quadratic cost function are only symmetric. Under a set of equivalent transformations, the indefinite LQ problem for rectangular DMJSs is equivalently turned into indefinite LQ problem for Markov jump systems (MJSs). On finite horizon, sufficient and necessary conditions are given for the solvability of the transformed indefinite LQ problem, and the exact optimal control and the optimal cost value are derived. Then, sufficient and necessary conditions are derived to guarantee that the transformed equivalent LQ problem for MJSs is definite, meanwhile, the unique optimal control and the non-negative optimal cost value are acquired. Besides, on infinite horizon, to ensure that the dynamic part for the optimal closed-loop system has a unique solution and is stochastically stable, several sufficient and necessary conditions are provided. Finally, two examples are presented as verifications of the theoretical results.