Open-Loop $H_{2}/H_{\infty }$ Control for Discrete-Time Mean-Field Stochastic Systems

This paper is concerned with the finite-horizon $H_{2}/H_{\infty }$ control problem about discrete-time mean-field linear stochastic systems with $(x,u,v)$ -multiplicative noises. We first present a mean-field stochastic bounded real lemma (MF-SBRL), which gives a necessary and sufficient condition for the linear perturbation operator of mean-field systems to gain an $H_{\infty }$ norm less than a prescribed disturbance attenuation level. Through a mean-field forward-backward stochastic difference equation (MF-FBSDE), an equivalent condition is proposed for the existence of open-loop $H_{2}/H_{\infty }$ control strategy. Based on the established MF-SBRL, it is further shown that the considered $H_{2}/H_{\infty }$ control problem is closed-loop solvable if and only if four coupled difference Riccati equations (CDREs) admit a set of positive semi-definite solutions. Moreover, the state-feedback gains of closed-loop $H_{2}/H_{\infty }$ control strategy are constructed in terms of the solutions to CDREs. As a by-product, the relationship is clarified between the open-loop solvability and the closed-loop solvability of finite-horizon mean-field $H_{2}/H_{\infty }$ control problem. Finally, a recursive algorithm appended with a numerical example is supplied to demonstrate that the adopted CDREs can be solved effectively.