Linear Quadratic Mean Field Game with Control Input Constraint

In this paper, we study a class of linear-quadratic (LQ) mean-field games in which the individual control process is constrained in a closed convex subset $\Gamma$ of full space $\mathbb{R}^m$. The decentralized strategies and consistency condition are represented by a class of mean-field forward-backward stochastic differential equation (MF-FBSDE) with projection operators on $\Gamma$. The wellposedness of consistency condition system is obtained using the monotonicity condition method. The related $\epsilon$-Nash equilibrium property is also verified.