During the past decade, variational problems formulated on infinite-dimensional approximations of multi-agent systems have become a central topic in applied mathematics. The main rationale behind the development of this field of study was - and still is - to provide efficient tools to investigate delicate dynamical properties for microsopic systems (e.g. self-organisation & competition, efficient control design, etc...) by means of suitable macroscopic approximations formulated in terms of mean-field limits.
In the context of mean-field optimal control, one aims at ensuring that control signals designed at the macroscopic level can be in turn used to stir the underlying class of microscopic systems. However, owing to the structure of the corresponding dynamics, which is modelled by a continuity equation, such a commutation is only possible under heavy regularity requirements on the driving fields. Moreover to this day, the only identified setting in which the non-local variants of continuity equations - which appear quasi systematically in multi-agent models - are known to be well-posed is that of Cauchy-Lipschitz regularity.
Motivated by these observations, I will present a work in collaboration with F. Rossi (Università degli Studi di Padova), in which we studied sufficient conditions ensuring that the optimal solution to optimal control problems in Wasserstein spaces are Lipschitz continuous with respect to the space variable. Our proof strategy is based on approximations by discrete optimal control problems, to which we carefully tailor recent results ensuring the existence of Lipschitz optimal feedbacks in finite dimensional optimal control problems. This combination of mean-field approximations and feedback synthesis, and in particular the uniformity of the corresponding estimates, relies crucially on a suitable use of the differential structures of Wasserstein spaces.