Discrete-time Mean-field Stochastic $H_2/H_\infty$ Control

Abstract: The finite horizon $H_2/H_\infty$ control problem of mean-field type for discrete-time systems is considered in this paper. Firstly, we derive a mean-field stochastic bounded real lemma (SBRL). Secondly, a sufficient condition for the solvability of discrete-time mean-field stochastic linear-quadratic (LQ) optimal control is presented. Thirdly, based on SBRL and LQ results, this paper establishes a sufficient condition for the existence of discrete-time stochastic $H_2/H_\infty$ control of mean-field type via the solvability of coupled matrix-valued equations.