In this paper, we consider a mean field game model inspired by crowd motion where agents aim to reach a closed set, called target set, in minimal time. Congestion phenomena are modeled through a constraint on the velocity of an agent that depends on the average density of agents around their position. The model is considered in the presence of state constraints: roughly speaking, these constraints may model walls, columns, fences, hedges, or other kinds of obstacles at the boundary of the domain which agents cannot cross. After providing a more detailed description of the model, the paper recalls some previous results on the existence of equilibria for such games and presents the main difficulties that arise due to the presence of state constraints. Our main contribution is to show that equilibria of the game satisfy a system of coupled partial differential equations, known mean field game system, thanks to recent techniques to characterize optimal controls in the presence of state constraints. These techniques not only allow to deal with state constraints but also require very few regularity assumptions on the dynamics of the agents.
Keywords: Mean field games, optimal control, minimal time, nonsmooth analysis, state constraints, MFG system, congestion games
@article{COCV_2022__28_1_A74_0,
author = {Sadeghi Arjmand, Saeed and Mazanti, Guilherme},
title = {Nonsmooth mean field games with state constraints},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2022},
publisher = {EDP-Sciences},
volume = {28},
doi = {10.1051/cocv/2022069},
mrnumber = {4524415},
zbl = {1505.49028},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2022069/}
}
TY - JOUR AU - Sadeghi Arjmand, Saeed AU - Mazanti, Guilherme TI - Nonsmooth mean field games with state constraints JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2022 VL - 28 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2022069/ DO - 10.1051/cocv/2022069 LA - en ID - COCV_2022__28_1_A74_0 ER -
%0 Journal Article %A Sadeghi Arjmand, Saeed %A Mazanti, Guilherme %T Nonsmooth mean field games with state constraints %J ESAIM: Control, Optimisation and Calculus of Variations %D 2022 %V 28 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2022069/ %R 10.1051/cocv/2022069 %G en %F COCV_2022__28_1_A74_0
Sadeghi Arjmand, Saeed; Mazanti, Guilherme. Nonsmooth mean field games with state constraints. ESAIM: Control, Optimisation and Calculus of Variations, Tome 28 (2022), article no. 74. doi: 10.1051/cocv/2022069
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G.M. was partially supported by ANR PIA funding: ANR-20-IDEES-0002.
