Heterogeneous Mean Field Social Control: The Multiplicative Noise Case

The linear quadratic mean field social control problem with heterogeneous agents and multiplicative noise is studied in this paper. The direct approach is adopted to design the decentralized controller for this large-population system. The heterogeneous agents are classified into different disjoint cluster, where the agents in a same cluster are homogeneous with identical dynamics and cost function. Agents are coupled with each other through the global cluster mean field term which is the stack of average state of agents in each cluster. First, under the centralized information pattern, we use variational analysis to obtain the necessary and sufficient conditions for the existence of optimal centralized open-loop controller characterized by a system of forward-backward stochastic differential equations (FBSDEs). Next, we decouple the high-dimensional FBSDEs with two coupled low-dimensional Riccati equations to obtain the optimal centralized feedback controller. Finally, under the decentralized information pattern, an asymptotically optimal decentralized controller is design by using the mean field approximations.