Optimal Control for Mean-field System: Discrete-time Case

This paper is concerned with discrete-time mean field linear-quadratic (LQ) control problem. A thorough solution to the problem is given for the first time. The sufficient and necessary condition for the solvability of mean-field LQ control problem is firstly presented in analytical expression based on the maximum principle developed in this paper, which is compared with the results obtained in literatures where only operator type solvability conditions were given. The optimal controller is given in terms of a coupled Riccati equation which is derived from the solution to forward and backward stochastic difference equation (FBSDE). The key techniques adopted in this paper are the maximum principle and the solution to the FBSDE obtained in this paper. The derived results in this paper will provide us the insight to solve the mean-field control problem for continuous-time systems and other related problems.