HUGHES MODEL: A MEAN FIELD GAMES APPROACH

This work aims to introduce Hughes' model and its properties. It is a model used to study pedestrian movement under certain theoretical assumptions. Our purpose is to investigate the fundamental properties of this model. We will do this by presenting Hughes' model based on real-world observations. Initially, we will derive a simple model based on the stationarity of the equations and the lack of viscosity. However, in Chapter 2, we will derive a new model to enable us to model various scenarios involving time. To achieve this, we will derive a Mean Field Games (MFG) system, starting from the derivation of MFGs from an optimal control problem. Indeed, MFG theory studies the evolution of strategic decisions made by players (represented as players' densities) aiming to minimize or maximize their payoff, expressed through a running cost over the duration of the game. The advantage of this MFG model lies in the many existence and uniqueness results for solutions. With this approach, we will be able to generalize Hughes' model, in the sense that, under suitable assumptions, we can recover some PDEs from the classical Hughes' model using the MFG system. Indeed, we would like to establish a set of suitable assumptions for our model to ensure its solvability. Moreover, this MFG system allows us to describe non-stationary conditions with viscosity effects. In the second part of this work, we will introduce a brief overview of stability analysis for the MFGs described above. Stability analysis studies how "stable" a solution to a PDEs system is. Stability relies on having "small" solutions with respect to initial data.