  This paper studies uniform stabilization and social optimality for linear
quadratic (LQ) mean field control problems with multiplicative noise, where
agents are coupled via dynamics and individual costs. The state and control
weights in cost functionals are not limited to be positive semi-definite. This
leads to an indefinite LQ mean field control problem, which may still be
well-posed due to deep nature of multiplicative noise. We first obtain a set of
forward-backward stochastic differential equations (FBSDEs) from variational
analysis, and construct a feedback control by decoupling the FBSDEs. By using
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 with
the help of linear matrix inequalities. A numerical example is given to
illustrate the effectiveness of the proposed control laws.
