A Mean Field Game System and a Related Deterministic Optimal Control Problem

TL;DR

This work analyzes a Mean Field Game system linked to Nash-type equilibria in large populations and shows the MFG system can be interpreted as the Euler–Lagrange equations of a deterministic optimal control problem constrained by a Fokker–Planck equation with drift control. It proves the existence of a weak solution to the MFG system and, under stronger convexity assumptions on the data, a uniqueness result; the approach relies on a fixed-point strategy over the density map induced by solving a linear FP and the associated Hamilton–Jacobi adjoint equation. The paper develops a robust variational-dynamic framework, including Yosida regularization, to handle the adjoint problem and ensure well-posedness beyond classical torus-domain settings. These results extend MFG well-posedness with weaker regularity, clarifying the deterministic optimal-control perspective and providing a foundation for further infinite-dimensional analysis and extensions.

Abstract

This paper concerns a Mean Field Game (MFG) system related to a Nash type equilibrium for dynamical games associated to large populations. One shows that the MFG system may be viewed as the Euler-Lagrange system for an optimal control problem related to a Fokker-Planck equation with control in the drift. One derives the existence of a weak solution to the MFG system and under more restrictive assumptions one proves a uniqueness result.