Section: New Results

Mathematical analysis and control of macroscopic traffic flow models

Vehicular traffic

Participants : Alessandra Cabassi, Maria Laura Delle Monache, Paola Goatin, Alexandre Bayen [UC Berkeley, CA, USA] , Legesse Lemecha Obsu [Addis Ababa University, Ethiopia] .

In collaboration with UC Berkeley, and as part of the Associated Team ORESTE activity (see http://www-sop.inria.fr/members/Paola.Goatin/ORESTE/index.html ), we have proposed a new junction model for ramp metering: we introduce a coupled PDE-ODE model, in which the PDE describes the evolution of the cars flow on the main lane 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. We were able to prove existence and uniqueness of the solution of the corresponding Riemann problem [41] . Relying on the above junction model, we have applied the Discrete Adjoint Method to efficiently compute (locally) optimal ramp-metering parameters to minimize the total travel time on a stretch of highway [80] .

In parallel, we have proposed two optimization strategy for instantaneous optimization of total travel times and total waiting times at roundabouts, which give an estimate of the time spent by drivers on the network section. These cost functionals are minimized with respect to the right-of-way parameter of the incoming roads. For each cost functional, the analytical expression is given for each junction, see [72] . This work is part of L.L. Obsu's PhD thesis.

Finally, we designed a new finite volume algorithm to track the trajectory of a bus in the surrounding traffic using a locally non-uniform moving mesh, see [3, 4, 5].

As part of our TRAM3 activity, we also organized the workshop “TRAM2 - Traffic Modeling and Management: Trends and Perspectives", which successfully took place at Inria Sophia Antipolis on March 20-22, 2013 (see https://team.inria.fr/opale/workshop-tram2/ ).

In the framework of the EIT ITC Labs Multimodal Mobility activity, A. Cabassi's internship was devoted to the calibration and the validation of a first order traffic flow model against processed real data provided by the industrial partners Autoroutes Traffic and VINCI Autoroutes, see [69] .

Crowd motion

Participants : Régis Duvigneau, Paola Goatin, Matthias Mimault, Debora Amadori [L'Aquila University, Italy] , Christophe Chalons [LJLL, UP7] , Massimiliano D. Rosini [ICM, Warsaw University, Poland] , Nicolas Seguin [LJLL, UPMC] , Monika Twarogowska.

From the analytical point of view, we have been studying the properties of some models in one space dimension. Concerning Hughes' scalar model, we have established a partial existence result in collaboration with D. Amadori and M.D. Rosini (see [75] ). M. Mimault's internship in 2012 was devoted to develop a MATLAB code based on wave-front tracking to compute the solutions of Hughes' model with generalized running cost, see [42] . He is currently working on a mixed hyperbolic-elliptic 2x2 system of conservation laws describing two groups of people moving in opposite directions. Finally, in collaboration with C. Chalons and N. Seguin, we generalized previous results on conservation laws with local flux constraints [3] , [5] to general flux functions and non-classical solutions arising in pedestrian flow modeling, see [39] . From the numerical point of view, we have implemented some macroscopic models in 2D on unstructured triangular meshes on the Num3sis platform. We provided a comparison between first and second order models in reproducing complex dynamics of crowd motion, such as formation of stop-and-go waves and clogging at bottlenecks. Then, we concentrated on the higher-order model and analyzed the dependence of the behavior of its solutions on some of the parameters of the system. In particular, we produced some examples where placing obstacles in front of the door prevents from blocking and decreases the evacuation time, see [73] , [81] .

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