Section: New Results

Mathematical analysis and control of macroscopic traffic flow models

Vehicular traffic

Participants : Maria Laura Delle Monache, Paola Goatin, Mauro Garavello [Piedmont University, Italy] .

Concerning road traffic, the research activity during 2011 focused on the mathematical analysis of traffic flow models on road networks or subject to unilateral constraints. In particular, [34] is devoted to a hyperbolic 2nd order model for traffic flow with local flux constraint. We describe two admissible Riemann solvers and we construct ad hoc finite volume numerical schemes to compute these solutions. The paper [59] is devoted to the study of a traffic flow model on a network composed by an arbitrary number of incoming and outgoing arcs connected together by a node with a buffer. We define the solution to the Riemann problem at the node and we prove existence and well posedness of solutions to the Cauchy problem. Finally, a general traffic flow model with phase-transition is proposed and described in [28] .

M.L. Delle Monache just started her doctoral thesis in the same topic. More precisely she will study hyperbolic models of traffic flow and associated optimization problems.

Crowd motion

Participants : Nora Aïssiouene, Régis Duvigneau, Nader El Khatib, Jihed Joobeur, Paola Goatin, Massimiliano D. Rosini [ICM, Warsaw University, Poland] .

Concerning pedestrian motion modeling, we are interested in the optimization of facilities design, in order to maximize pedestrian flow and avoid or limit accidents due to panic situations. To this aim, we are now studying a macroscopic model for crowd movements consisting in a scalar conservation law accounting for mass conservation coupled with an Eikonal equation giving the flux direction depending on the density distribution. From the theoretical point of view, and as a first step, we are studying the problem in one space dimension (for applications, this case corresponds to a crowd moving in a corridor). In collaboration with M.  Rosini (supported by the project CROM3, funded by the PHC Polonium 2011), we have established entropy conditions to select physically relevant solutions, and we have constructed explicit solutions for some simple initial data (these results are presented in [54] ). We are now studying existence and uniqueness of solutions of the corresponding initial boundary value problem. From the numerical point of view, we are implementing the model in two space dimensions on triangular meshes on the Num3sis platform. This was partly done by N. El-Khatib (postdoc at INRIA from January to August 2011), and will be completed soon by Nora Aïssiouene. This will provide a performing numerical tool to solve the related optimization problems arising in the optimization of facilities design, such as the position and size of an obstacle in front of (before) a building exit in order to maximize the outflow through the door and avoid or limit over-compression. Moreover, jointly with the PULSAR team, we have supervised J. Joobeur's internship, which was devoted to pedestrian data collection from real-word video recordings (Turin metro station). The density data will serve to validate the model.

The above researches were partially funded by the ERC Starting Grant "TRAM3 - Traffic management by macroscopic models".