Discrete Time Mean-Field Stochastic Linear-Quadratic Optimal Control Problems

摘要

This paper first presents necessary and sufficient conditions for thesolvability of discrete time, mean-field, stochastic linear-quadratic optimalcontrol problems. Then, by introducing several sequences of bounded linearoperators, the problem becomes an operator stochastic LQ problem, in which theoptimal control is a linear state feedback. Furthermore, from the form of theoptimal control, the problem changes to a matrix dynamic optimization problem.Solving this optimization problem, we obtain the optimal feedback gain and thusthe optimal control. Finally, by completing the square, the optimality of theabove control is validated.