论文标题
网络上的一阶平均现场游戏
First order Mean Field Games on networks
论文作者
论文摘要
本文致力于有限的地平线确定性平均野外游戏，其中状态空间是网络。代理控制其速度，当他们占据顶点时，他们可以进入任何事件边缘。假定运行和终端成本在每个边缘都连续，但不一定在网络上全球连续。提出和研究了拉格朗日的配方。它导致松弛的平衡，包括对可允许轨迹的概率措施。获得了这种松弛的平衡的存在。证明要求存在最佳轨迹和地图的封闭图属性，该图与每个点相关联的最佳轨迹集。对于任何放松的平衡，都对应于平均野外游戏的温和解决方案，即由相关最佳控制问题的价值函数$ u制成的一对$（u，m）$，以及一个家庭$ m =（m（t））_ t _ t _ t _ t $概率度量。给定$ m $，值函数$ u $的特征是网络上的汉密尔顿 - 雅各布问题。研究了$ U $的规律性物业和$ m $满足的Fokker-Planck方程的薄弱形式。
This paper is devoted to finite horizon deterministic mean field games in which the state space is a network. The agents control their velocity, and when they occupy a vertex, they can enter into any incident edge. The running and terminal costs are assumed to be continuous in each edge but not necessarily globally continuous on the network. A Lagrangian formulation is proposed and studied. It leads to relaxed equilibria consisting of probability measures on admissible trajectories. The existence of such relaxed equilibria is obtained. The proof requires the existence of optimal trajectories and a closed graph property for the map which associates to each point the set of optimal trajectories starting from that point. To any relaxed equilibrium corresponds a mild solution of the mean field game, i.e. a pair $(u,m)$ made of the value function $u$ of a related optimal control problem, and a family $m= (m(t))_t$ of probability measures on the network. Given $m$, the value function $u$ is characterized by a Hamilton-Jacobi problem on the network. Regularity properties of $u$ and a weak form of a Fokker-Planck equation satisfied by $m$ are investigated.
