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.
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