First order mean field games in crowd dynamics

In this paper we study a two dimensional crowd model where pedestrian velocity consists of two elements: a non--local interaction term, modeling the effect of other walkers on each individual, and a control term. This latter term can be chosen by pedestrians so that their resulting path is optimal w.r.t. a suitable cost criterion. Under the assumption that pedestrians can forecast the effect of their choices on the evolution of the whole crowd, it is natural to consider a mean field system coupling the continuity equation, which describes the evolution of the density of pedestrians, with the HJB equation for the optimization problem. We show that such coupled system admits solution and we interpret the solution in terms of the pedestrian models. We also extend this results to the case where multiple populations of pedestrians are present in the environment, each with his own objectives.