Lqg Optimal Control Arising in Mean Field Decision Problems

Abstract. We consider a linear-quadratic-Gaussian (LQG) optimal control problem where the generalized state space is the product of an Euclidian space and an infinite dimensional function space. This model originates from a mean field LQG game with a major player and a large number of minor players, and has importance in designing decentralized strategies in the game. We show that the underlying infinite dimensional stochastic system has a well defined solution for each admissible control, and there is a unique optimal control. Subsequently, we introduce an adjoint equation system parametrized by the infinite parameter set and determine the linear optimal control. Finally, by introducing infinite dimensional Riccati-like equations we obtain Markovian linear optimal control.