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Section: New Results

Mathematical analysis and control of macroscopic traffic flow models

Vehicular traffic

Participants : Guillaume Costeseque, Paola Goatin, Christophe Chalons [UVST] , Simone Göttlich [U Mannheim, Germany] , Jerôme Härri [EURECOM] , Oliver Kolb [U Mannheim, Germany] , Sosina Mengistu-Gashaw [EURECOM] , Francesco Rossi [U Aix-Marseille] , Stefano Villa [U Milano-Bicocca] .

The thesis of S. Mengistu-Gashaw, funded by the Labex UCN@Sophia (http://ucnlab.eu/ ) and co-supervised by P. Goatin and J. Härri, is devoted to understanding and modeling mobility characteristics of scooters and motorcycles for user-centric ITS application. We are currently developing a macroscopic model for heterogeneous traffic including car and motorcycles.

A new traffic flow model has been designed in [44] for taking into account the multiclass and multilane features of real traffic. This model is based on a system of coupled Hamilton-Jacobi PDEs for an appropriate choice of framework that mixes spatial and Lagrangian coordinates. The coupling conditions emerge from the moving bottleneck theory that has been developed in the traffic flow literature several years ago but for which a real mathematical sound basis lacked. Very recently, there were some new results dealing with the existence of a solution under suitable assumptions. However, these results were set for the hyperbolic conservation law in Eulerian coordinates and they are not straightforward to extend to Hamilton-Jacobi equations in different coordinates. Despite that the well-posedness of the problem is still an open problem, a numerical method is developed and it takes advantage of the classical representation formula available for HJ PDEs. This numerical scheme has been proved to provide good qualitative results.

In collaboration with F. Rossi, we proved existence and uniqueness of solutions to a transport equation modelling vehicular traffic in which the velocity field depends non-locally on the downstream traffic density via a discontinuous anisotropic kernel. The result is obtained recasting the problem in the space of probability measures equipped with the $\infty$-Wasserstein distance. We also show convergence of solutions of a finite dimensional system, which provide a particle method to approximate the solutions to the original problem. See, [45] .

Finally, the internship of S. Villa, co-supervised by M. Garavello (U Milano-Bicocca), was devoted to the analytical and numerical study of the Aw-Rascle-Zhang model with moving bottleneck. Two Riemann Solver have been proposed, and two numerical strategies have been developed. A journal article is in preparation in collaboration with C. Chalons.

Crowd motion

Participants : Paola Goatin, Matthias Mimault.

M. Mimault defended hid PhD on December 14th, 2015. The last part of his thesis was devoted to the numerical study of scalar conservation laws with non-local flow in two space dimensions. These equations are meant to model crowd motion, where the movement direction of each pedestrian depends on a weighted mean of the crowd density around him. In particular, he implemented a finite volume numerical scheme which has been used for flow optimization purposes: he applied the adjoint method to compute the gradient for the evacuation time minimization depending on the initial crowd distribution.