Stability Analysis of a Non-Separable Mean-Field Games for Pedestrian Flow in Large Corridors

We investigate a generalized Hughes model for pedestrian flow in an infinitely large corridor. We demonstrate that constant flows are stable under a condition on the density. This model is a non-separable mean-field game, which has a backward/forward structure, and the questions of existence and stability of small perturbations of constant states were previously unanswered. Our findings suggest that when the density is below half of the maximal density \textit{$\dfrac{\rho_{m}}{2}$}, we can control the perturbation and prove stability results for the nonlinear Generalized Hughes model. However, due to wave propagation phenomena, we are unable to provide an answer about the stability results when the density is higher. Our approach involves constructing an explicit solution for the linear problem in Fourier analysis and demonstrating, through a fixed-point argument, how to construct the solution for the full nonlinear mean-field games system.