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）的公式通常要求哈密顿量具有连续可微性，以便确定玩家密度的Kolmogorov-Fokker-Planck方程中的对流项。然而，在许多实际应用中，底层最优控制问题可能呈现bang-bang控制，这通常导致不可微的哈密顿量。本文针对凸的、Lipschitz连续但可能不可微的哈密顿量的一般情况，发展了稳态MFG的分析与数值分析。特别地，我们基于凸函数次微分来理解哈密顿量的导数，将MFG系统推广为偏微分包含（PDI）。我们证明了MFG PDI系统弱解的存在性，并进一步在类似于Lasry和Lions考虑的单调性条件下证明了唯一性。随后，我们提出了该问题的单调有限元离散化，并证明了价值函数近似在强$H^1$范数下的收敛性以及密度函数近似在强$L^q$范数下的收敛性。我们通过数值实验展示了该数值方法在处理非光滑解时的性能。