Differential inclusions in Wasserstein spaces: The Cauchy-Lipschitz framework
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 whos...
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| Published in | Journal of Differential Equations Vol. 271; pp. 594 - 637 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
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15.01.2021
Elsevier |
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| 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. |
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| AbstractList | 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. |
| Author | Frankowska, Hélène Bonnet, Benoît |
| Author_xml | – sequence: 1 givenname: Benoît orcidid: 0000-0002-0893-5324 surname: Bonnet fullname: Bonnet, Benoît email: benoit.bonnet@imj-prg.fr – sequence: 2 givenname: Hélène surname: Frankowska fullname: Frankowska, Hélène email: helene.frankowska@imj-prg.fr |
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| Keywords | 28B20 49J45 Relaxation and compactness of trajectories 34G20 49J21 Continuity equation Mean-field optimal control Differential inclusion 34A60 Optimal transport Filippov theorem Relaxation Optimal Transport Filippov Theorem Com- pactness of Trajectories Mean-Field Optimal Control MSC2010 Subject Classification : 28B20 Differential Inclusion Continuity Equation |
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| SubjectTerms | Analysis of PDEs Continuity equation Differential inclusion Filippov theorem Mathematics Mean-field optimal control Optimal transport Optimization and Control Probability Relaxation and compactness of trajectories |
| Title | Differential inclusions in Wasserstein spaces: The Cauchy-Lipschitz framework |
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