Differential inclusions in Wasserstein spaces: The Cauchy-Lipschitz framework

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:271

Article

Bonnet, Benoit¹  Frankowska, Helene¹ 

[1] Sorbonne Univ, CNRS, IMJ PRG, UMR 7586, 4 Pl Jussieu, F-75252 Paris, France

关键词: Continuity equation;    Differential inclusion;    Optimal transport;    Filippov theorem;    Relaxation and compactness of trajectories;    Mean-field optimal control;   

DOI  :  10.1016/j.jde.2020.08.031

来源: Elsevier

【 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. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free