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Finite horizon stochastic $H_2/H_\infty$ control for continuous-time mean-field systems with Poisson jumps

The stochastic $H_2/H_\infty$ control problem for continuous-time mean-field stochastic differential equations with Poisson jumps over finite horizon is investigated in this paper. Continuous and jump diffusion terms in the system depend not only on the state but also on the control input, external disturbance, and mean-field components. By employing the quasi-linear technique and the method of completing the square, a mean-field stochastic jump bounded real lemma of the system is derived, which plays a crucial role in solving stochastic $H_2/H_\infty$ control problem. It is demonstrated in this study that the feasibility of the stochastic $H_2/H_\infty$ control problem is equivalent to the solvability of four sets of cross-coupled generalized differential Riccati equations, thus generalizing the previous results to mean-field jump-diffusion systems. To validate the proposed methodology, a numerical simulation example is provided to illustrate the effectiveness of the control strategy. The results establish a systematic approach for designing $H_2/H_\infty$ controllers that simultaneously guarantee the robustness against disturbances and optimal performance for interacting particle systems.