Generalized Coupled Algebraic Riccati Equations for Discrete-time Markov Jump with Multiplicative Noise Systems

In this paper we consider the existence of the maximal and mean square stabilizing solutions for a set of generalized coupled algebraic Riccati equations (GCARE for short) associated to the infinite-horizon stochastic optimal control problem of discrete-time Markov jump with multiplicative noise linear systems. The weighting matrices of the state and control for the quadratic part are allowed to be indefinite. We present a sufficient condition, based only on some positive semidefinite and kernel restrictions on some matrices, under which there exists the maximal solution and a necessary and sufficient condition under which there exists the mean square stabilizing solution for the GCARE. We also present a solution for the discounted and long run average cost problems when the performance criterion is assumed to be composed by a linear combination of an indefinite quadratic part and a linear part in the state and control variables. The paper is concluded with a numerical example for pension fund with regime switching.