PMP-based numerical solution for mean field game problem of general nonlinear system

This brief paper investigates a numerical algorithm to derive the optimal solution for the nonlinear mean field game problem. Previously, the linear quadratic Gaussian problem was explored deeply since the analytical solution for it can be derived without employing the Fokker–Planck (FP) equation. However, for the general mean field game problem, the FP equation must be employed to describe the mean field dynamics of all the agents. In this brief paper, the optimal condition is obtained through Pontryagin’s maximum principle first; then, this mean field dynamics described by the partial differential equation is calculated through the Lax–Friedrichs approach. Finally, the initial co-state in the optimal condition is determined by the Newton method. Two case studies, including vehicle speed consensus and engine speed control of HEVs with consideration of powertrain structure limitation, are conducted, and the simulation results in both practical applications show the effectiveness of the proposed algorithm.