Mild and weak solutions of Mean Field Games problem for linear control systems
Mild and weak solutions of Mean Field Games problem for linear control systems
复制标题
线性控制系统平均场游戏问题的弱解和弱解
DOI：
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发表时间：
2019
期刊：
影响因子：
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通讯作者：
Cristian Mendico
中科院分区：
文献类型：
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作者：
P. Cannarsa;Cristian Mendico
本文研究了$mathbb{R}^{d}$上线性受控动态下的一阶平均场对策。对于这类问题，我们定义纳什均衡（称为平均场博弈均衡）作为Borel概率测度的空间上的可允许的轨迹，我们证明了这样的平衡的存在性和唯一性。此外，我们还研究了温和解的正则性：我们证明了平均场对策平衡点的保持器正则性和最优控制问题的值函数的分数阶凹腔。最后，我们提出了与平均场博弈问题相关的偏微分方程系统，并证明了温和解类与弱解类是一致的。
The aim of this paper is to study first order Mean field games subject to a linear controlled dynamics on $mathbb{R}^{d}$. For this kind of problems, we define Nash equilibria (called Mean Field Games equilibria) as Borel probability measures on the space of admissible trajectories and we prove the existence and uniqueness of such equilibria. Moreover, we study the regularity of mild solutions: we prove Holder regularity of Mean Field Games equilibria and fractional semiconcavity for the value function of the underlying optimal control problem. In conclusion, we presents the PDEs system associated with the Mean Field Games problem and we prove that the class of mild solutions coincide with the class of weak solutions.