On the stochastic linear quadratic optimal control problem by piecewise constant controls: The infinite horizon time case

This paper is devoted to the problem of indefinite stochastic linear quadratic (LQ) optimal control by piecewise constant controls in an infinite horizon case. By restricting the set of admissible controls to the class of piecewise constant stochastic processes, we reformulated the above control problem under the setting of systems modeled by Ito differential equations controlled by impulses. We show that the solution in a state feedback form of the indefinite stochastic LQ control problem is equivalent to the existence of a global stabilizing solution associated to a class of backward matrix linear differential equations with a Riccati type jumping operators.