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] , Alexandre Bayen [UC Berkeley, CA, USA] .

The activity in traffic flow modeling has being reinforced by the creation of the Associated Team ORESTE between OPALE and the UC Berkeley teams Mobile Millennium and Integrated Corridor Management (ICM) lead by Prof. A. Bayen (see http://www-sop.inria.fr/members/Paola.Goatin/ORESTE/index.html ). In this framework, three PhD students from US visited Inria during August and September, and M.L. Delle Monache spent two and half months at UC Berkeley.

During this first year of common research we proposed a new junction model for ramp-metering in the continuous and discrete settings. We focused on a junction consisting in a mainline, an on-ramp and an off-ramp. In particular, we introduced a coupled PDE-ODE model, in which the PDE describes the evolution of the cars flow on the mainline and the ODE describes the evolution of the queue length on the on-ramp, modeled by a buffer, which ensures that boundary conditions are satisfied in strong sense. At the junction we imposed the maximization of the outgoing flux together with a fixed priority parameter for incoming roads. We were able to prove existence and uniqueness of the solution of the corresponding Riemann problem. This approach has then been extended to networks and discretized using the Godunov scheme. The corresponding discrete optimization problem has been solved using the Adjoint Method and it is now being implemented into a MATLAB code. This model will serve as starting point for a subsequent model for optimal rerouting, which includes multi-commodity flow and partial control.

Besides that, we studied a a coupled PDE-ODE system modeling the interaction of a large slow moving vehicle with the surrounding traffic flow. The model consists in a scalar conservation law with moving density constraint describing traffic evolution coupled with an ODE for the slow vehicle trajectory. The constraint location moves due to the surrounding traffic conditions, which in turn are affected by the presence of the slower vehicle, thus resulting in a strong non-trivial coupling. The existence result is given in [60] .

The paper [41] 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.

Crowd motion

Participants : Nora Aïssiouene, Christophe Chalons [LJLL, UP7] , Régis Duvigneau, Paola Goatin, Matthias Mimault, Massimiliano D. Rosini [ICM, Warsaw University, Poland] , Nicolas Seguin [LJLL, UPMC] , Monika Twarogowska.

The activity on in pedestrian flow modeling is reinforced by the doctoral thesis of M. Mimault, started in October, and the enrollment of M. Twagorowska on a post-doctoral position.

Concerning crowd 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 first and second order macroscopic models for crowd movements consisting in one or two scalar conservation law accounting for mass conservation and momentum balance, 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 [40] ). We are now studying existence of solutions of the corresponding initial boundary value problem, using the wave-front tracking approach. In this framework, M. Mimault's internship was devoted to develop a Matlab code based on wave-front tracking to compute the solutions of Hughes' model of pedestrian motion with generalized running cost. This model displays a non-classical dynamic at the splitting point between the two directions of motion. The wave-front tracking scheme provides us with reference solutions to test numerically the convergence of classical finite volume schemes, which do not treat explicitly the dynamics at the turning point (see [66] ). The code can be downloaded at the following URL: http://www-sop.inria.fr/members/Paola.Goatin/wft.html

From the numerical point of view, we are implementing some macroscopic models 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 is now being completed by M. Twarogowska, with the support of N. 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.

Finally, in collaboration with C. Chalons and N. Seguin, we have generalized the results on conservation laws with local flux constraint obtained in [3] , [5] to general flux functions and nonclassical solutions arising for example in pedestrian flow modeling. We first define the constrained Riemann solver and the entropy condition, which singles out the unique admissible solution. We provide a well posedness result based on wave-front tracking approximations and Kruzhkov doubling of variable technique. We then provide the framework to deal with nonclassical solutions and we propose a “front-tracking” finite volume scheme allowing to sharply capture classical and nonclassical discontinuities. Numerical simulations illustrating the Braess paradox are presented as validation of the method. The results are collected in [65] .

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