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First order Mean Field Games with density constraints: Pressure equals Price

Cardaliaguet, Pierre; Mészáros, Alpár Richárd; Santambrogio, Filippo (2016), First order Mean Field Games with density constraints: Pressure equals Price, SIAM Journal on Control and Optimization, 54, 5, p. 2672-2709. 10.1137/15M1029849

Type
Article accepté pour publication ou publié
External document link
https://hal.archives-ouvertes.fr/hal-01173947
Date
2016
Journal name
SIAM Journal on Control and Optimization
Volume
54
Number
5
Pages
2672-2709
Publication identifier
Metadata
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Author(s)
Cardaliaguet, Pierre
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Mészáros, Alpár Richárd
UCLA Department of mathematics
Santambrogio, Filippo
Laboratoire de Mathématiques d'Orsay [LM-Orsay]
Abstract (EN)
In this paper we study Mean Field Game systems under density constraints as optimality conditions of two optimization problems in duality. A weak solution of the system contains an extra term, an additional price imposed on the saturated zones. We show that this price corresponds to the pressure field from the models of incompressible Euler's equations à la Brenier. By this observation we manage to obtain a minimal regularity, which allows to write optimality conditions at the level of single agent trajectories and to define a weak notion of Nash equilibrium for our model.
Subjects / Keywords
Mean Field Game systems