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Intrinsic Lipschitz regularity of mean-field optimal controls. (English) Zbl 1466.35068

Summary: In this article, we provide sufficient conditions under which the controlled vector fields solution of optimal control problems formulated on continuity equations are Lipschitz regular in space. Our approach involves a novel combination of mean-field approximations for infinite-dimensional multi-agent optimal control problems, along with a careful extension of an existence result of locally optimal Lipschitz feedbacks. The latter is based on the reformulation of a coercivity estimate in the language of Wasserstein calculus, which is used to obtain uniform Lipschitz bounds along sequences of approximations by empirical measures.

MSC:

35B65 Smoothness and regularity of solutions to PDEs
49J20 Existence theories for optimal control problems involving partial differential equations
49J30 Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
58E25 Applications of variational problems to control theory
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