Static output feedback strategy for mean-field social control with nonlinear stochastic dynamics

A mean-field social control problem for uncertain nonlinear stochastic systems is investigated by using a robust static output feedback (SOF) strategy. First, the problem in the single decision maker case is investigated in terms of guaranteed cost control approaches to derive suboptimal conditions at the supremum of the cost function. The Karush-Kuhn-Tucker (KKT) condition is used to derive the necessary conditions which are expressed as a large stochastic combined matrix equation (SCME). Second, the preliminary results in the single decision maker case are used to study the Pareto optimal strategy in a cooperative game. As our main contribution, we derive the high-order centralised strategies and the low-order decentralised strategies, respectively, for the cooperative game. In order to avoid the difficulty of higher-order dimensional problem related to SCMEs, a new reduced-order decomposition numerical scheme by means of Newton's method is developed. The computation for designing the proposed strategy set can be performed in low dimension, even when the number of decision makers approachs to infinity. Moreover, the degradation of the cost function is rigorously evaluated by comparing the centralised strategy set with the proposed strategy set. Finally, several numerical experiments are conducted to demonstrate the usefulness and effectiveness of the proposed strategy set.