Infinite Horizon Stabilization and Linear Quadratic Optimal Control of Descriptor Stochastic Markov Jump Systems

The linear quadratic (LQ) optimal control problem with indefinite weighting matrices and the stabilization problem for discrete-time descriptor stochastic Markov jump systems (DSMJSs) involving state-dependent noises are studied. By using the Moore-Penrose generalized inverse of matrices and the equivalent transformation of restricted system, the indefinite LQ problem for DSMJSs is equivalently converted into the indefinite LQ problem for Markov jump systems (MJSs). Under some viable conditions and stabilization assumption, the generalized stochastic algebraic Riccati equation having a unique semi-positive definite solution is guaranteed. Then the necessary and sufficient conditions which ensure that DSMJSs are causal and mean-square stable are established. It is shown that the admissibility of the optimal closed-loop systems is equivalent to stabilizability in mean-square sense of the transformed MJSs. Besides, an efficient iterative algorithm is given to verify the mean-square stabilizability of DSMJSs by solving an optimization problem. Two examples including a practical RLC circuit system are presented as verifications of the theoretical results.