论文标题
Wasserstein空间中的差异夹杂物：Cauchy-Lipschitz框架
Differential Inclusions in Wasserstein Spaces: The Cauchy-Lipschitz Framework
论文作者
论文摘要
在本文中，我们提出了一个一般框架，以研究概率措施的瓦斯汀空间中的差异夹杂物。基于对连续性方程结构的较早几何见解，我们将差分包含物的解定义为绝对连续曲线的驱动速度场是可测量的多功能选择，以在矢量场的空间中占据其值。在这种一般环境中，我们证明了差分理论的三个基础：Filippov的定理，放松定理和溶液集的紧凑性。这些贡献基于对连续性方程解决方案的新估计的新贡献，然后应用于通过闭环控制的完全非线性平均均值最佳控制问题获得新的存在结果。
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.
