Non smooth evolution models in crowd dynamics: mathematical and numerical issues

Abstract Accounting for hard congestion in crowd motion mod-eling leads to non-smooth evolution problems. At the microscopic level (individuals are represented separately), these problems fit in the framework of non smooth analysis in Hilbert spaces, and the tools developed in the 70’s to handle the so-called sweeping pro-cess are directly adaptable. At the macroscopic scale (the popu-lation is represented by a density), a similar approach can be car-ried out. This is done by identifying densities with measures in the Wasserstein space, endowed with the distance based on optimal transportation for the quadratic cost. These lecture notes provide an introduction to the mathematical theory of these models, and a description of numerical methods, in the microscopic (ODE) and macroscopic (PDE) cases.