NECESSARY AND SUFFICIENT CONDITIONS OF OPEN-LOOP AND CLOSED-LOOP SOLVABILITY FOR DELAYED STOCHASTIC LQ OPTIMAL CONTROL PROBLEMS

In this paper, a linear quadratic optimal control problem driven by a stochastic differential delay system is investigated, where both state delay and control delay can appear in the state equation, especially in the diffusion term. Three kinds of solvability for the delayed control problem are proposed---open-loop solvability, the closed-loop representation of open-loop optimal control, and closed-loop solvability---and their necessary and sufficient conditions are obtained. The delayed control problem is transformed into an infinite dimensional optimal control problem without delay but with a new control operator. Some novel auxiliary equations are constructed to overcome the difficulties caused by the new control operator, because state delay and control delay coexist, and some stochastic analysis tools are lacking in the study of the above three kinds of solvability. The open-loop solvability is ensured by the solvability of a constrained forward-backward stochastic evolution system and a convexity condition, or by the solvability of an anticipated-backward stochastic differential delay system and a convexity condition; the closed-loop representation of the open-loop optimal control is given via a coupled matrix-valued Riccati equation; the closed-loop solvability is ensured by the solvability of an operator-valued Riccati equation or a coupled matrix-valued Riccati equation.