Modeling crowd dynamics by the mean-field limit approach

We propose to couple through optimal control theory, a dynamical crowd evolution models where pedestrians make decisions with a well-defined set of actions and strategies by minimizing their utility function. We explain how situations of this kind can be modeled by making use of differential game theory. By leveraging on some simple examples, we introduce the most fundamental concept of mean field games recently introduced by Lasry and Lions [J.-M. Lasry, P.-L. Lions, Mean field games, Jpn. J. Math. 2 (1) (2007) 229–260]. The main interest of mean field games in our work is to simplify the interactions between pedestrians. As a consequence, we derive a continuum formulation of the crowd dynamics and nonlinear systems involving partial differential equations of crowd models.