Optimal Stabilization Control for Discrete-Time Mean-Field Stochastic Systems

This paper will investigate the stabilization and optimal linear quadratic (LQ) control problems for infinite horizon discrete-time mean-field systems. Unlike the previous works, for the first time, the necessary and sufficient stabilization conditions are explored under mild conditions, and the optimal LQ controller for infinite horizon is designed with a coupled algebraic Riccati equation (ARE). More specifically, we show that under the exact detectability (exact observability) assumption, the mean-field system is stabilizable in the mean square sense with the optimal controller if and only if a coupled ARE has a unique positive semidefinite (positive definite) solution. The presented results are parallel to the classical results for the standard LQ control.