Abstract

We present a problem described by Mean Field Games (MFG) and Optimal Control theory on finite time horizon. This problem consists of a system of PDEs: a Kolmogorov–Fokker–Planck equation, evolving forward in time and a Hamilton–Jacobi–Bellman equation, evolving backwards in time. The numerical difficulties are based on a turnpike effect considered in this paper. We present an extremal problem whose necessary conditions of extremal satisfy the initial system of PDEs, and introduce its numerical solution at the heart of monotonic schemes. According to special assumptions, PDEs can be reduced to Riccati ODEs. We consider this reduction as a test example for the numerical solution of the extremal problem.

中文翻译：

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收费公路平均场博弈问题的数值解

摘要

我们提出了在有限时间范围内用均值博弈（MFG）和最优控制理论描述的问题。这个问题由一系列PDE组成：一个Kolmogorov–Fokker–Planck方程，随时间向前发展；一个Hamilton–Jacobi–Bellman方程，随时间向后发展。数值上的困难是基于本文考虑的收费尖峰效应。我们提出一个极值问题，其极值的必要条件满足PDE的初始系统，并将其数值解引入单调方案的核心。根据特殊假设，PDE可以还原为Riccati ODE。我们认为这种减少是极值问题数值解的测试示例。