Necessary and Sufficient Solution to Optimal Control for Linear Continuous Time Mean-field System

Abstract

Different from previous works, this paper will provide the necessary and sufficient (essential) solution to linear quadratic (LQ) optimal control problem for continuous-time mean-field systems. Firstly by applying the Maximum Principle developed in this paper, the necessary and sufficient solvability condition of the finite horizon mean-field optimal control problem is presented in an explicit expression form (in analytical form) for the first time, which is completely different from the operator-type conditions obtained in previous works. Secondly, we explore the analytical solution to the forward and backward stochastic differential equation (FBSDE) of the mean-field optimal control problem. Accordingly, the optimal controller, in terms of system state and its expectation, is designed via a coupled Riccati equation. It is very interesting to show that the coupled Riccati equation is derived from the solution to the FBSDE, which has the similarity with the case of standard optimal control.