Differential inclusions in Wasserstein spaces: the Cauchy-Lipschitz framework. (English) Zbl 1454.49018
Authors’ 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.
Reviewer: Andrej V. Plotnikov (Odessa)
MSC:
| 49J45 | Methods involving semicontinuity and convergence; relaxation |
| 49J21 | Existence theories for optimal control problems involving relations other than differential equations |
| 28B20 | Set-valued set functions and measures; integration of set-valued functions; measurable selections |
| 34A60 | Ordinary differential inclusions |
| 34G20 | Nonlinear differential equations in abstract spaces |
| 49Q22 | Optimal transportation |
| 60B05 | Probability measures on topological spaces |
Keywords:
continuity equation; differential inclusion; optimal transport; Filippov theorem; relaxation and compactness of trajectories; mean-field optimal controlReferences:
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