一维平均场对策规划的势方法
A Potential Approach for Planning Mean-Field Games in One Dimension
摘要：
本文讨论了一阶和二阶一维平均场博弈（MFG）的规划问题。这些游戏是由一个哈密尔顿-雅可比方程耦合福克-普朗克方程。将庞加莱引理应用于Fokker-Planck方程，导出了势的存在性.将Hamilton-Jacobi方程用位势改写，得到了一类变分问题的Euler-Lagrange方程组。代替平均场规划问题（MFP），我们研究这个变分问题。利用变分学中的直接法，证明了该变分问题解的存在唯一性。变分法具有消除连续性方程的优点。我们还考虑了一阶MFP与拥塞。通过引入一个势函数并借助变分不等式理论证明了该拥塞问题存在弱解。最后，我们提出了一个应用程序的一维休斯模型的文件。
英文摘要：
This manuscript discusses planning problems for first- and second-order one-dimensional mean-field games (MFGs). These games are comprised of a Hamilton-Jacobi equation coupled with a Fokker-Planck equation. Applying Poincaré's Lemma to the Fokker-Planck equation, we deduce the existence of a potential. Rewriting the Hamilton-Jacobi equation in terms of the potential, we obtain a system of Euler-Lagrange equations for certain variational problems. Instead of the mean-field planning problem (MFP), we study this variational problem. By the direct method in the calculus of variations, we prove the existence and uniqueness of solutions to the variational problem. The variational approach has the advantage of eliminating the continuity equation. We also consider a first-order MFP with congestion. We prove that the congestion problem has a weak solution by introducing a potential and relying on the theory of variational inequalities. We end the paper by presenting an application to the one-dimensional Hughes' model.
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