Section: New Results

Mathematical analysis and control of macroscopic traffic flow models

Vehicular traffic

Participants : Enrico Bertino, Guillaume Costeseque, Maria Laura Delle Monache, Paola Goatin, Sheila Scialanga, Alexandre Bayen [UC Berkeley, CA, USA] , Sebastien Blandin [IBM Research Collaboratory, Singapore] , Christophe Chalons [LJLL, UP7] .

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 considered the System Optimal Dynamic Traffic Assignment problem with Partial Control (SO-DTA-PC) for general road networks with horizontal queuing. The goal of which is to optimally control any subset of the networks agents to minimize the total congestion of all agents in the network. We adopt a flow dynamics model that is a Godunov discretization of the Lighthill-Williams-Richards (LWR) partial differential equation with a triangular flux function and a corresponding multi-commodity junction solver. Full Lagrangian paths are assumed to be known for the controllable agents, while we only assume knowledge of the aggregate turn ratios for the non-controllable (selfish) agents. We solve the resulting finite horizon non-linear optimal control problem using the discrete adjoint method, see [75] .

As part of our TRAM3 activity and in collaboration with C. Chalons (UVSQ), we designed a new finite volume conservative algorithm to track the trajectory of a bus in the surrounding traffic using a locally non-uniform moving mesh, see [70] .

In collaboration with S. Blandin (IBM), we proved the existence and stability of entropy weak solutions of a scalar conservation law with non-local flux arising in traffic flow modeling. The result is obtained providing accurate $L^{\infty }$, $BV$ and $L^1$ estimates for the sequence of approximate solutions constructed by an adapted Lax-Friedrichs scheme.

In collaboration with the University of Mannheim and in the framework of the PHC Procope project “Transport Networks Modeling and Analysis”, we studied how to manage variable speed limits combined with coordinated ramp metering within the framework of the LWR network model. Following a "first discretize then optimize" approach, we derived the first order optimality system and explained the switch of speeds at certain fixed points in time and the boundary control for the ramp metering as well. Sequential quadratic programming methods are used to solve the control problem numerically. For application purposes, we present experimental setups where variable speed limits are used as a traffic guidance system to avoid traffic jams on highway interchanges and on-ramps, see [71] .

Finally, E. Bertino internship was devoted to uncertainty quantification in macroscopic traffic flow models.

Crowd motion

Participants : Aekta Aggarwal, Régis Duvigneau, Paola Goatin, Matthias Mimault, Rinaldo M. Colombo [Brescia University, Italy] .

A. Aggarwal postdoc is devoted to the analytical and numerical study of systems of conservation laws with non-local fluxes in several space dimensions. In collaboration with R.M. Colombo, we presented a Lax-Friedrichs type algorithm to numerically integrate this class of systems. The convergence of the approximate solutions was proved, also providing the existence of solution in a slightly more general setting than in other results in the current literature. An application to a crowd dynamics model is considered. This numerical algorithm is then used to test the conjecture that as the convolution kernels converge to a Dirac $\delta$, the nonlocal problem converges to its non-nonlocal analogue.

M. Mimault is working on 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, M. Mimault is implementing a finite volume numerical scheme which will be used for flow optimization purposes.

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