论文标题
固定二阶平均野外游戏部分差分夹杂物的分析和数值近似
Analysis and Numerical Approximation of Stationary Second-Order Mean Field Game Partial Differential Inclusions
论文作者
论文摘要
平均野外游戏（MFG）的表述通常需要对哈密顿量的连续可不同，以确定Kolmogorov中的对流术语 - Fokker--Planck方程的玩家密度。但是，在许多具有实际兴趣的情况下，潜在的最佳控制问题可能表现出爆炸控制，这通常会导致无差的哈密顿人。我们为凸，Lipschitz的一般情况（但可能是非不同的哈密顿人）的固定MFG进行了分析和数值分析。特别是，我们建议将MFG系统作为部分差分包含（PDI），基于解释凸函数的亚差异的衍生物。我们确定了MFG PDI系统的弱解决方案，并且在与Lasry和Lions所考虑的单调性条件类似的单调条件下进一步证明了唯一性。然后，我们提出了问题的单调有限元离散化，并证明了近似值的强$ h^1 $ - norm收敛，并强$ l^q $ - norm融合了密度函数的近似值。我们说明了具有非平滑溶液的数值实验中数值方法的性能。
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.
