4.6 Article

INTRINSIC LIPSCHITZ REGULARITY OF MEAN-FIELD OPTIMAL CONTROLS

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Differential inclusions in Wasserstein spaces: The Cauchy-Lipschitz framework

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Summary: In this paper, a general framework for studying differential inclusions in the Wasserstein space of probability measures is proposed, with solutions defined as absolutely continuous curves driven by measurable selection functions taking values in the space of vector fields. The foundational results of Filippov's theorem, Relaxation theorem, and compactness of solution sets are proved in this setting, based on novel estimates on solutions of continuity equations. These contributions 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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