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Theses

Modeling traffic on urban road networks with hyperbolic conservation laws

Abstract : This thesis is devoted to the modeling of traffic flow using hyperbolic conservation laws, with a specific focus on urban applications. Urban areas are today facing severe episodes of air pollution and increasing congestion due to traffic. The objective is to overcome some of the current limitations of macroscopic traffic flow models in urban situations. We first study the seminal Aw-Rascle-Zhang model with relaxation. We prove well-posedness of the model using wave-front tracking approximations and splitting technique in a Lagrangian setting. Besides, we provide an estimate on the decay of positive waves. We then show that the solutions of the Aw-Rascle-Zhang system with relaxation converge to a weak solution of the LWR model when the relaxation parameter goes to zero. Finally, we propose a discussion on the entropy aspect of this weak solution of the LWR model. We then propose a new macroscopic traffic flow model accounting for the boundedness of traffic acceleration, which is required for physical realism. Our model is built on the coupling between the scalar conservation law accounting for the conservation of vehicles and a number of ordinary differential equations describing the trajectories of accelerating vehicles, which we treat as moving constraints. We detail a wave-front tracking algorithm to construct approximate solutions of the model, with general flux functions and show existence of solutions to the Cauchy problem for a piecewise constant initial datum. Finally, we provide numerical simulations of the model in different urban situations, from a single Riemann problem to sequences of traffic lights, and confront the results to numerical simulations of the LWR model. Finally, we introduce a new macroscopic traffic flow model with buffers on road networks. This model features buffers of finite size, enabling backward propagation of congestion on the network, and time-dependent routing functions at the junctions. The dynamics are first defined on the level of conservation laws, and then transformed in an Hamilton-Jacobi formulation. We prove existence, uniqueness and stability of the solutions with respect to the routing ratios and initial datum using a fixed-point problem in a proper Banach space. Thanks to stability, the model provides a controllable framework, using routing ratios as control parameters. This represents an advance towards solving the Dynamic Traffic Assignment (DTA) problem. In the end we detail how this framework applies to a classical road network with several intersections and finite-length links.
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Nicolas Laurent-Brouty. Modeling traffic on urban road networks with hyperbolic conservation laws. Analysis of PDEs [math.AP]. Université Côte d'Azur, 2019. English. ⟨NNT : 2019AZUR4056⟩. ⟨tel-02488804v2⟩

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