A simple model to handle the flow of people in emergency evacuation situations is considered: at every point x, the velocity U (x) that individuals at x would like to realize is given. Yet, the incompressibility constraint prevents this velocity field to be realized and the actual velocity is the projection of the desired one onto the set of admissible velocities. Instead of looking at a microscopic setting (where individuals are represented by rigid discs), here the macroscopic approach is investigated, where the unknown is a density ρ(t,x). If a gradient structure is given, say U = -∇D where D is, for instance, the distance to the exit door, the problem is presented as a Gradient Flow in the Wasserstein space of probability measures. The functional which gives the Gradient Flow is neither finitely valued (since it takes into account the constraints on the density), nor geodesically convex, which requires for an ad hoc study of the convergence of a discrete scheme.
Optimality conditions for a controlled sweeping process with applications to the crowd motion model
