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Traffic Management by Macroscopic Models

Final Report Summary - TRAM3 (Traffic Management by Macroscopic Models)

Project TRAM3 focused on the study of macroscopic models for vehicular and pedestrian traffic, and how optimal control approaches can be used in traffic management. Due to unconventional models and the presence of shock waves in the solutions, standard techniques do not apply for establishing well-posedness results and for solving control and optimization problems.
The scientific achievements obtained by the project addressed road traffic modeling on networks, the minimization of associated cost functionals (such as the total travel time) and the analytical and numerical study of crowd motion models and their optimal control. In particular, we have obtained original results concerning:
- Scalar conservation laws with moving density constraints arising in the modeling of moving bottlenecks due to buses or trucks interacting with the surrounding traffic: This results in a strongly coupled PDE-ODE system, for which we have provided an existence result and two numerical strategies to compute approximate solutions.
- Efficient numerical strategies to compute optimal ramp-metering and re-routing controls to minimize total travel times on road networks. This required the design of suitable junction models, and was based on an efficient use of the Discrete Adjoint Method to compute cost gradients.
- Well-posedness and convergence of micro-macro limits for a new traffic model with velocity depending non-locally on a weighted mean of the downstream density: this model satisfies the maximum principle and displays finite acceleration.
- The analytical study of a first order macroscopic model for pedestrian motion in one space dimension, for which we proposed an admissibility condition on weak solutions. Moreover, we implemented a wave-front tracking scheme providing reference solutions to test numerically the convergence of classical finite volume schemes, which are not specifically designed for this type of problems. We also provided an existence result for a mixed hyperbolic-elliptic 2x2 system of conservation laws describing two groups of people moving in opposite directions.
- Comparative numerical study of first and second order models of pedestrian motion (in two space dimensions). In particular, we observed that first order models are incapable of reproducing complex dynamics of crowd motion at bottlenecks, such as the formation of stop-and-go waves and clogging. As a consequence, they are not suitable to the study of shape optimization problems arising in pedestrian facilities design. To this purpose, second order model should be privileged, for which we produced some examples where placing obstacles in front of the door prevents from blocking and decreases the evacuation time.
- General conservation laws with local flux constraints arising in pedestrian flow modeling: We provided the framework to deal with non-classical solutions and we propose a “front-tracking” finite volume scheme allowing to sharply capture classical and non-classical discontinuities.
- Non-local models in several space dimensions, providing existence of solutions by proving convergence of a Lax-Friedrichs type numerical scheme. The results were applied to the implementation of a discrete adjoint method to compute the optimal initial distribution of pedestrians in a room to minimize the evacuation time.