Infinite-Time Minimal Cost Variance Control and Coupled Algebraic Riccati Equations

Minimum cost variance control (MCV) optimizes the variance of the cost function while the cost mean is kept at a prespecified level. The solutions of the infinite time horizon full-state-feedback MCV problem are found using the Hamilton-Jacobi theory. In the solutions of infinite time horizon MCV control problem, a pair of coupled algebraic Riccati equations arises. This paper considers the existence of a positive semidefinite solution pair for the steady-state version of coupled algebraic Riccati equations, where one entry of the pair corresponds to cost mean, while the other entry of the pair corresponds to cost variance. For the MCV control problem, existence and uniqueness of the solutions of the coupled algebraic Riccati equations are provided. From this result it is established that the MCV feedback controller stabilizes the closed loop system. Furthermore, the algorithm to find the MCV controller from the coupled algebraic Riccati equations is presented. Finally, three examples are provided to verify the existence theorem of the coupled algebraic Riccati equations.