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Mild and weak solutions of Mean Field Games problem for linear control systems

Cannarsa, Piermarco; Mendico, Cristian (2020), Mild and weak solutions of Mean Field Games problem for linear control systems, Minimax Theory and its Applications, 5, 2, p. 221-250

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1907.02654.pdf (278.1Kb)
Type
Article accepté pour publication ou publié
Date
2020
Journal name
Minimax Theory and its Applications
Volume
5
Number
2
Publisher
Heldermann Verlag
Pages
221-250
Metadata
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Author(s)
Cannarsa, Piermarco
Dipartimento di Matematica [Roma II] [DIPMAT]
Mendico, Cristian
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Gran Sasso Science Institute [GSSI]
Abstract (EN)
The aim of this paper is to study first order Mean field games subject to a linear controlled dynamics on Rd. For this kind of problems, we define Nash equilibria (called Mean Field Games equilibria), as Borel probability measures on the space of admissible trajectories, and mild solutions as solutions associated with such equilibria. Moreover, we prove the existence and uniqueness of mild solutions and we study their regularity: we prove Hölder regularity of Mean Field Games equilibria and fractional semiconcavity for the value function of the underlying optimal control problem. Finally, we address the PDEs system associated with the Mean Field Games problem and we prove that the class of mild solutions coincides with a suitable class of weak solutions.
Subjects / Keywords
Mean field games; mean field games equilibrium; semiconcave estimates; control systems

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