Indefinite Linear Quadratic Mean Field Social Control Problems With Multiplicative Noise

This article 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 semidefinite. 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 virtue 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 with the help of linear matrix inequalities. A numerical example is given to illustrate the effectiveness of the proposed control laws.