Calculus of Variations and Geometric Measure Theory
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A. R. Mészáros

Mean Field Games with density constraints

created by mészáros on 05 Oct 2015


M2 Mémoire/MSc thesis

Inserted: 5 oct 2015
Last Updated: 5 oct 2015

Year: 2012

École Polytechnique, advisor: Filippo Santambrogio


This work is devoted to the study of some MFG models (introduced by J.-M. Lasry and P.-L. Lions) with an additional constraint that the density of the agents should remain below a certain given threshold. This question was initiated by F. Santambrogio and motivated by some macroscopic crowd motion models under hard congestion effects studied by him and his coauthors.

The thesis contains the used tools from the theory of optimal transport and some well-known properties of Hamilton-Jacobi equation arising in from the theory of optimal control of ODEs. It describes the crowd motion models and the classical MFG theory in a nutshell as well. An attempt to prove the existence of a solution of a first order MFG system under the density constraint is presented. In addition, the thesis contains a uniqueness result for crowd motion models under density constraint, obtained by the contractive property of the 2-Wasserstein distance along two solutions.


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