Finite and infinite clusters mean field control problems via graphon theory

In this paper, we study the social optimality for mean field linear-quadratic control systems following direct method, where subsystems are coupled via individual dynamics and costs. A graph is introduced to represent the large-population system, where nodes represent subpopulations called clusters and edges represent coupling relationship. First, a gauge transformation is used to decouple the mean field system in the finite clusters case then the optimal controller under the centralized information pattern is obtained. Based on centralized results and graphs, distributed and decentralized controller are further designed by mean field approximations, and the asymptotically social optimality of the decentralized controller is further proved. Finally, the situation of infinite clusters is studied through graphon theory. The results show that two Riccati equations are needed, one of which is a low-dimensional calculated by its own subpopulation information and a high-dimensional calculated by all population information.