Semi-Discrete Shocks for a Microscopic Pedestrian Model

Abstract

In this thesis, we explore in the framework of viscosity solutions the existence and uniqueness of traveling solutions using a pedestrian microscopic model. We consider that the velocity of the pedestrian at position u(y) depends on the velocity of two pedestrians ahead. u′(y) = V (u(y + 1) − u(y)) + V (u(y + 2) − u(y)) Throughout this work, we give the definition of a viscosity solution first then we prove the exponential behavior of the solution at ±∞. After that, using the monotony of the pedestrians interdistance which we achieve by the strong comparison principle, we derive necessarily conditions for the existence of such solutions. With this established, we proceed by constructing a traveling solution considering an approximate non-local operator on a bounded domain and using Perron’s method.