A Class of Non Local Models for Pedestrian Traffic Rinaldo M Colombo1 Mauro Garavello2 Magali Lecureux Mercier3

Niveau: Secondaire
A Class of Non-Local Models for Pedestrian Traffic Rinaldo M. Colombo1, Mauro Garavello2, Magali Lecureux-Mercier3 April 5, 2011 Abstract We present a new class of macroscopic models for pedestrian flows. Each individual is assumed to move towards a fixed target, deviating from the best path according to the instantaneous crowd distribution. The resulting equation is a conservation law with a nonlocal flux. Each equation in this class generates a Lipschitz semigroup of solutions and is stable with respect to the functions and parameters defining it. Moreover, key qualita- tive properties such as the boundedness of the crowd density are proved. Specific models are presented and their qualitative properties are shown through numerical integrations. In particular, the present models account for the possibility of reducing the evacuation time from a room by carefully positioning obstacles that direct the crowd flow. 2000 Mathematics Subject Classification: 35L65, 90B20. Keywords: Crowd Dynamics, Macroscopic Pedestrian Model, Non-Local Conservation Laws. 1 Introduction From a macroscopic point of view, a moving crowd is described by its density ? = ?(t, x), so that for any subset A of the plane, the quantity ∫ A ?(t, x) dx is the total number of individ- uals in A at time t. In standard situations, the number of individuals is constant, so that conservation laws of the type ∂t? + divx (?v) = 0 are the natural tool for the description of crowd dynamics.

• numerical integrations


• rn ?


• datum ?0 ?


• crowd dynamics


• ?r1 ?


• crowd density


• time through


• analytical proofs


• path choice