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189 people work at LJLL

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January 2022


Guilherme Mazanti

Monday 23 September 2019

Guilherme Mazanti (Université Paris-Sud)
Minimal-time mean field games.

Abstract :
Mean field games (MFGs) have been extensively studied since their introduction around 2006 by the independent works of P. E. Caines, M. Huang, and R. P. Malhamé and J.-M. Lasry and P.-L. Lions. Such differential games with a continuum of indistinguishable agents have been proposed as approximations of games with a large number of symmetric players, finding their applications in several domains. Motivated by the problem of proposing MFG models for crowd motion, this talk considers a MFG where agents want to leave a bounded domain through a part of its boundary in minimal time, an agent’s maximal speed being bounded in terms of the distribution of agents around their position in order to model congestion phenomena.

After presenting the model and its motivation, we establish the existence of equilibria for this game using a fixed-point strategy based on a Lagrangian formulation. In the case where agents may leave the domain through any point of its boundary, thanks to some further regularity properties of optimal trajectories obtained through Pontryagin’s maximum principle, we characterize equilibria through a MFG system. We also present some simulations in simple situations and discuss other recent results and ongoing work, including sufficient conditions ensuring the L^p regularity of the distribution of agents and the case where agents’ dynamics are stochastic.

This talk is based on joint works with Samer Dweik and Filippo Santambrogio.