On the Planning Problem for the Mean Field Games System
On the Planning Problem for the Mean Field Games System
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平均场游戏系统的规划问题
DOI：
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发表时间：
2014
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影响因子：
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通讯作者：
A. Porretta
中科院分区：
文献类型：
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作者：
A. Porretta
我们考虑一类平均场博弈的规划问题，包括价值函数 u 的 Hamilton-Jacobi-Bellman 方程和玩家密度 m 的 Fokker-Planck 方程的耦合系统，而人们希望通过代理的最优决策，将玩家密度从给定的初始配置驱动到时间 T 的目标密度。假设成本准则中的耦合 F(x,m) 相对于 m 是单调的，并且哈密顿量在二次函数的上下边界上有一定的增长，我们证明了系统存在一个弱解，对于密度 m 具有规定的初始和终止条件 m0、m1（正且平滑）。这也是福克-普朗克方程通过某些最优输运场精确可控性结果的特例。
We consider the planning problem for a class of mean field games, consisting in a coupled system of a Hamilton–Jacobi–Bellman equation for the value function u and a Fokker–Planck equation for the density m of the players, whereas one wishes to drive the density of players from the given initial configuration to a target one at time T through the optimal decisions of the agents. Assuming that the coupling F(x,m) in the cost criterion is monotone with respect to m, and that the Hamiltonian has some growth bounded below and above by quadratic functions, we prove the existence of a weak solution to the system with prescribed initial and terminal conditions m0, m1 (positive and smooth) for the density m. This is also a special case of an exact controllability result for the Fokker–Planck equation through some optimal transport field.