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.

翻译：平均场博弈（MFG）的经典公式通常要求哈密顿函数具有连续可微性，以确定描述玩家密度的Kolmogorov-Fokker-Planck方程中的对流项。然而，在许多实际问题中，底层最优控制问题可能涉及bang-bang控制，这通常导致哈密顿函数不可微。我们针对凸、Lipschitz但可能不可微的哈密顿函数这一一般情形，发展了平稳MFG的分析与数值分析。特别地，我们基于凸函数的次微分概念来诠释哈密顿函数的导数，提出将MFG系统推广为偏微分包含（PDI）形式。我们证明了MFG PDI系统弱解的存在性，并在与Lasry和Lions所考虑的单调性条件类似的条件下进一步证明了解的唯一性。随后，我们提出该问题的单调有限元离散格式，并证明了值函数近似解在强$H^1$范数下的收敛性以及密度函数近似解在强$L^q$范数下的收敛性。最后，通过涉及非光滑解的数值实验展示了该数值方法的性能。