SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 138-166, February 2024.
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 the 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 [math]-norm convergence of the approximations of the value function and strong [math]-norm convergence of the approximations of the density function. We illustrate the performance of the numerical method in numerical experiments featuring nonsmooth solutions.

中文翻译：

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平稳二阶平均场博弈偏微分包含分析与数值逼近

SIAM 数值分析杂志，第 62 卷，第 1 期，第 138-166 页，2024 年 2 月。
摘要。平均场博弈 (MFG) 的制定通常需要哈密顿量的连续可微性，以便确定 Kolmogorov-Fokker-Planck 方程中玩家密度的平流项。然而，在许多具有实际意义的情况下，潜在的最优控制问题可能会表现出爆炸式控制，这通常会导致不可微的哈密顿量。我们针对凸、Lipschitz 的一般情况（但可能是不可微的哈密顿量）开发了平稳 MFG 的分析和数值分析。特别是，我们基于用凸函数的次微分解释哈密顿量的导数，提出将 MFG 系统推广为偏微分包含（PDI）。我们建立了 MFG PDI 系统弱解的存在性，并进一步证明了在与 Lasry 和 Lions 所考虑的类似单调性条件下的唯一性。然后，我们提出了问题的单调有限元离散化，并证明了值函数近似的强[数学]范数收敛性和密度函数近似的强[数学]范数收敛性。我们在具有非光滑解的数值实验中说明了数值方法的性能。