Social Optima in Linear Quadratic Mean-field Output Feedback Control

This paper studies social optimal control problem for linear quadratic mean-field system with noisy output. For the output feedback control, each agent needs to estimate not only its own state but also the population average state, where both states are coupled with each other. Firstly, to deal with the noisy output, we give the optimal estimator of the state in the form of the sum of two Kalman filters. Then, by variational method, a set of forward-backward stochastic difference equations (FBSDEs) are presented, and output feedback-type centralized controllers are acquired from the backward iterative method by decoupling the FBSDEs. Finally, by mean field approximations, we further design a set of decentralized controllers, which is proved to have asymptotic social optimality.