Abstract 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.

中文翻译：

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Wasserstein 空间中的微分包含：Cauchy-Lipschitz 框架

摘要 在本文中，我们提出了一个通用框架，用于研究 Wasserstein 概率测度空间中的微分包含。基于对连续性方程结构的早期几何见解，我们将微分包含的解定义为绝对连续的曲线，其驱动速度场是在矢量场空间中取值的多功能的可测量选择。在这个一般设置中，我们证明了微分包含理论的三个创始结果：Filippov 定理、松弛定理和解集的紧致性。这些贡献——基于对连续性方程解的新估计——然后被应用于推导出具有闭环控制的完全非线性平均场最优控制问题的新存在结果。