Wasserstein空间中的微分包含：Cauchy-Lipschitz框架
Differential Inclusions in Wasserstein Spaces: The Cauchy-Lipschitz Framework
摘要：
在这篇文章中，我们提出了一个一般的框架研究概率测度的Wasserstein空间中的微分包含。基于对连续性方程结构的早期几何认识，我们将微分包含的解定义为绝对连续的曲线，其驱动速度场是取其值于向量场空间的多功能函数的可测选择。在这个一般的设置，我们证明了三个创始结果的理论微分包含：Filippov定理，松弛定理，和紧的解决方案集。这些贡献-这是基于新的估计的连续性方程的解决方案-然后应用到推导出一个新的存在性结果完全非线性平均场最优控制问题的闭环控制。
英文摘要：
In this article, we propose a general framework for the study of differential inclusions in the Wasserstein space of probability measures. Based on earlier geometric insights on the structure of continuity equations, we define solutions of differential inclusions as absolutely continuous curves whose driving velocity fields are measurable selections of multifunction taking their values in the space of vector fields. In this general setting, we prove three of the founding results of the theory of differential inclusions: Filippov's theorem, the Relaxation theorem, and the compactness of the solution sets. These contributions -- which are based on novel estimates on solutions of continuity equations -- are then applied to derive a new existence result for fully non-linear mean-field optimal control problems with closed-loop controls.
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