Linear Quadratic Optimal Control for Discrete-time Mean-field Systems with Input Delay

The linear quadratic (LQ) optimal control and stabilization problems for mean-field systems with input delay (MFSID) are investigated in this paper. The necessary and sufficient solvability conditions for LQ control of MFSID are first given in terms of a convexity condition and the solvability of equilibrium conditions. Consequently, by solving the associated mean-field forward and backward stochastic difference equations (MF-FBSDEs), the optimal control is derived in terms of the solution of a modified Riccati equation (MRE). Furthermore, for infinite horizon case, the stabilization problem for MFSID is studied, and the necessary and sufficient stabilizability conditions are obtained. We show that MFSID can be mean square stabilizable if and only if a modified algebraic Riccati equation (MARE) admits a unique positive definite solution.