Asymptotically Optimal Distributed Control for Linear–Quadratic Mean Field Social Systems With Heterogeneous Agents

The linear–quadratic mean field social control problem is studied in a large-population system with heterogeneous agents following the direct approach. A graph is introduced to represent the network topology of the large-population system, where nodes represent subpopulations called clusters and edges represent communication relationship. Agents in each cluster are homogeneous and 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, by tackling high-dimensional FBSDEs with two coupled low-dimensional Riccati equations, we construct the optimal centralized feedback controller, which composed of the individual state and the global cluster mean field term. Then, under the distributed information pattern, an asymptotically unbiased mean field estimator for each agent is designed to estimate the local cluster mean field terms according to the given network topology. Finally, a set of asymptotically optimal distributed controllers is proposed based on the mean field estimator and its asymptotically social optimality is further proved. A numerical simulation is conducted to demonstrate the effectiveness of the proposed distributed controller.