*M2 Mémoire/MSc thesis*

**Inserted:** 5 oct 2015

**Last Updated:** 5 oct 2015

**Year:** 2012

**Notes:**

École Polytechnique, advisor: Filippo Santambrogio

**Abstract:**

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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