Analysis and Numerical Approximation of Stationary Second-Order Mean Field Game Partial Differential Inclusions

TL;DR

This paper extends the analysis and numerical approximation of stationary Mean Field Games to cases with nondifferentiable Hamiltonians by formulating a Partial Differential Inclusion, establishing existence, uniqueness, and convergence of solutions.

Contribution

It introduces a novel PDI framework for stationary MFGs with nonsmooth Hamiltonians, proving existence, uniqueness, and developing a convergent finite element discretization.

Findings

Established existence of weak solutions for MFG PDIs.

Proved convergence of the numerical scheme in relevant norms.

Demonstrated effectiveness on problems with nonsmooth solutions.

Abstract

The formulation of Mean Field Games (MFG) typically requires continuous differentiability of the Hamiltonian in order to determine the advective term in the Kolmogorov--Fokker--Planck equation for the density of players. However, in many cases of practical interest, the underlying optimal control problem may exhibit bang-bang controls, which typically lead to nondifferentiable Hamiltonians. We develop the analysis and numerical analysis of stationary MFG for the general case of convex, Lipschitz, but possibly nondifferentiable Hamiltonians. In particular, we propose a generalization of the MFG system as a Partial Differential Inclusion (PDI) based on interpreting the derivative of the Hamiltonian in terms of subdifferentials of convex functions. We establish existence of a weak solution to the MFG PDI system, and we further prove uniqueness under a similar monotonicity condition to the one considered by Lasry and Lions. We then propose a monotone finite element discretization of the problem, and we prove strong $H^1$-norm convergence of the approximations to the value function and strong $L^q$-norm convergence of the approximations of the density function. We illustrate the performance of the numerical method in numerical experiments featuring nonsmooth solutions.