Optimal control of linear discrete-time systems with multiplicative noises

This paper studies mean-square stabilization and optimal control problems via state feedback for linear discrete-time systems with state and control multiplicative noises. We first show that in general the state feedback stabilization problem in mean-square sense amounts to solving a generalized eigenvalue problem(GEVP). Next, we pose the H₂ optimal control problem equivalent to an optimal mean-square stabilization problem. As a consequence, both the mean-square stabilization and the H₂ optimal control problems can be solved efficiently as one of generalized eigenvalue problems, for which computational algorithms are readily available. The optimal state feedback in turn can be designed by solving a modified algebraic Riccati equation (MARE).