RESUMEN
This article focuses on the mean-field linear-quadratic Pareto (MF-LQP) optimal strategy design for stochastic systems in infinite horizon, which is with the H∞ constraint when the system is disturbed by external interferences. The stochastic bounded real lemma (SBRL) with any initial state in infinite horizon is first investigated based on the stabilizing solution of the generalized algebraic Riccati equation (GARE). Then, by discussing the convexity of the cost functional, the stochastic indefinite MF-LQP control problem is defined and solved based on the MF-LQ theory and Pareto theory. When the worst case disturbance is considered in the collaborative multiplayer system, we show that the Pareto optimal strategy design with H∞ constraint [or robust Pareto optimal strategy, (RPOS)] can be given via solving two coupled GAREs. When the worst case disturbance and the Pareto efficient strategy work, all Pareto solutions are obtained by a generalized Lyapunov equation. Finally, a practical example shows that the obtained results are effective.
RESUMEN
This article presents results on designing the Pareto-optimal strategy under H∞ constraint for the linear mean-field stochastic systems disturbed by external disturbances. First, combining the stochastic H∞ 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∞ 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.