IJCAS

This paper is concerned with the optimal control and stabilization problems for discrete-time Markov jump systems with multi-channel multiplicative noises. The weight matrices in the cost function can be indefinite. The necessary and sufficient existence conditions for the optimal controller in the finite horizon is proposed explicitly in terms of generalized difference Riccati equations. Further, via the finite horizon optimal cost, we define a Lyapunov function to reduce the indefinite stabilization problem to a definite one. And in the infinite horizon, the necessary and sufficient condition to stabilize the Markov jump linear system in the mean-square sense is presented. Moreover, we prove that the Markov jump linear system is mean-square stabilizable if and only if the generalized coupled algebraic Riccati equation has a solution. Simultaneously, this solution is the maximal solution to the generalized coupled algebraic Riccati equations.

Keywords: Markov jump linear system, multiplicative noise, optimal control, Riccati equation, stabilization.