Abstract

This paper studies social optima for linear quadratic mean field control systems with multiplicative noises, where the state and control weights in costs are not limited to be semi-positive definite. We first obtain a set of forward-backward stochastic differential equations (FBSDEs) from variational analysis, and construct a feedback-type control by decoupling the FBSDEs. With the help of solutions to two Riccati equations, we design a set of decentralized control laws, which is further shown to be asymptotically social optimal. Some equivalent conditions are given for uniform stabilization of the systems.

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