Linear Quadratic Mean Field Social Control with Random Coefficients and Common Noise

This paper studies uniform stabilization and social optimality for linear quadratic mean field control systems with common noise, where the coefficients in individual dynamics and costs are random and time-varying. We first obtain a set of forward-backward stochastic differential equations (FBSDEs) from variational analysis, and construct a centralized feedback control by decoupling the FBSDEs. By using solutions of two backward stochastic Riccati equations, we design a set of decentralized controls, which is further shown to be asymptotically social optimal. The necessary and sufficient conditions are given for uniform stabilization of the systems by exploiting the relation between population state average and aggregate effect.