Pareto-Optimal Strategy for Linear Mean-Field Stochastic Systems With H∞ Constraint

This article presents results on designing the Pareto-optimal strategy under $H_{\infty }$ constraint for the linear mean-field stochastic systems disturbed by external disturbances. First, combining the stochastic $H_{\infty }$ control theory with the stochastic mean-field theory, we derive the stochastic bounded real lemma (SBRL) of our considered linear mean-field stochastic systems with the stochastic initial condition. Second, we use the mean-field forward–backward stochastic differential equation to solve the mean-field linear quadratic Pareto-optimal problem with indefinite cost functionals. It is proved that the existence of a closed-loop Pareto-optimal strategy is equivalent to the solvability of the coupled generalized differential Riccati equations when some conditions are satisfied. Finally, a necessary and sufficient condition for the Pareto-optimal strategy under the $H_{\infty }$ constraint is researched by four-coupled matrix-valued equations. Besides, we also obtain the Pareto frontier for the mean-field stochastic system with only state-dependent noise. A practical example is presented to show the effectiveness of our main results.