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SUMMARY:Sufficient conditions for the Lipschitz regularity of mean-field o
ptimal controls
DTSTART;VALUE=DATE-TIME:20210329T080000Z
DTEND;VALUE=DATE-TIME:20210329T090000Z
DTSTAMP;VALUE=DATE-TIME:20210508T092717Z
UID:indico-event-6447@indico.math.cnrs.fr
DESCRIPTION:During the past decade\, variational problems formulated on in
finite-dimensional approximations of multi-agent systems have become a cen
tral topic in applied mathematics. The main rationale behind the developme
nt 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 me
ans of suitable macroscopic approximations formulated in terms of mean-fie
ld limits.\n\nIn the context of mean-field optimal control\, one aims at e
nsuring that control signals designed at the macroscopic level can be in t
urn used to stir the underlying class of microscopic systems. However\, ow
ing to the structure of the corresponding dynamics\, which is modelled by
a continuity equation\, such a commutation is only possible under heavy re
gularity requirements on the driving fields. Moreover to this day\, the on
ly identified setting in which the non-local variants of continuity equati
ons - which appear quasi systematically in multi-agent models - are known
to be well-posed is that of Cauchy-Lipschitz regularity.\n\nMotivated by t
hese observations\, I will present a work in collaboration with F. Rossi (
UniversitÃ degli Studi di Padova)\, in which we studied sufficient condit
ions ensuring that the optimal solution to optimal control problems in Was
serstein spaces are Lipschitz continuous with respect to the space variabl
e. Our proof strategy is based on approximations by discrete optimal contr
ol problems\, to which we carefully tailor recent results ensuring the exi
stence of Lipschitz optimal feedbacks in finite dimensional optimal contro
l problems. This combination of mean-field approximations and feedback syn
thesis\, and in particular the uniformity of the corresponding estimates\,
relies crucially on a suitable use of the differential structures of Wass
erstein spaces.\n\nhttps://indico.math.cnrs.fr/event/6447/
LOCATION:Ã€ distance
URL:https://indico.math.cnrs.fr/event/6447/
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