Indefinite Stochastic Linear Quadratic Control in Infinite Time Horizon

In this paper, we will be concerned with an optimal stochastic linear-quadratic (LQ in short) control problem in an infinite time horizon, where the cost matrices are allowed to be indefinite. For which, a generalized algebraic Riccati equation (CARE in short) which involves a matrix pseudoinverse is introduced. Under some stabilizing conditions, we establish the relation between the solvability of the GARE and the existence of the optimal controls of the LQ control problem. Based on this relation, the optimal controls can be constructed by the solution of the GARE if it exists. Finally, we give a sufficient condition for the solvability of the GARE based on the semidefinite programming.