The standard formulation of the PDE system of Mean Field Games (MFG) requires the differentiability of the Hamiltonian. However in many cases, the structure of the underlying optimal problem leads to a convex but non-differentiable Hamiltonian. For time-dependent MFG systems, we introduce a generalization of the problem as a Partial Differential Inclusions (PDI) by interpreting the derivative of the Hamiltonian in terms of the subdifferential set. In particular, we prove the existence and uniqueness of weak solutions to the resulting MFG PDI system under standard assumptions in the literature. We propose a monotone stabilized finite element discretization of the problem, using conforming affine elements in space and an implicit Euler discretization in time with mass-lumping. We prove the strong convergence in $L^2(H^1)$ of the value function approximations, and strong convergence in $L^p(L^2)$ of the density function approximations, together with strong $L^2$-convergence of the value function approximations at the initial time.

翻译：标准形式的平均场博弈（MFG）偏微分方程组要求Hamilton量可微。然而在许多情况下，底层最优问题的结构会导致凸但不可微的Hamilton量。针对依赖于时间的MFG系统，我们通过将Hamilton量的导数解释为次微分集合，引入该问题的一个推广形式——偏微分包含（PDI）。特别地，我们在文献中的标准假设下证明了所得MFG PDI系统弱解的存在唯一性。我们提出该问题的单调稳定化有限元离散格式，采用空间上的协调仿射单元和时间上的隐式欧拉离散结合质量集中。我们证明了值函数逼近在$L^2(H^1)$中的强收敛性，密度函数逼近在$L^p(L^2)$中的强收敛性，以及初始时刻值函数逼近的强$L^2$收敛性。