Homogenization of Hamilton-Jacobi equations and applications to road traffic.

Summary This thesis contains two contributions to the space-time homogenization of the first order Hamilton-Jacobi equations. These equations are related to the modelling of road traffic. Finally, some results of homogenization in a nearly periodic environment are presented. The first chapter is devoted to the homogenization of an infinite system of coupled differential equations with delay time. This system is derived from a microscopic model of simple road traffic. The drivers follow each other on an infinite straight road and their reaction time is taken into account. The speed of each driver is assumed to be a function of the distance to the preceding driver: we speak of a "follow-the-leader" model. Thanks to a strict comparison principle, we show the convergence to a macroscopic model for reaction times lower than a critical value. In a second step, we show a counterexample to the homogenization for a reaction time higher than this critical value, for particular initial conditions. For this purpose, we perturb the stationary solution in which the vehicles are all equidistant at the initial times. The second chapter deals with the homogenization of a Hamilton-Jacobi equation whose Hamiltonian is spatially discontinuous. The associated traffic model is a straight road with an infinite number of traffic lights. The traffic lights are assumed to be identical, equally spaced and the time delay between two successive lights is assumed to be constant. We study the large-scale influence of this phasing on the traffic. We distinguish between the free road portion, which will be represented by a macroscopic model, and the traffic lights, which will be modeled by time-dependent flow limiters. The theoretical framework is the one by C. Imbert and R. Monneau (2017) for Hamilton-Jacobi equations on networks. The study consists in the theoretical homogenization, where the effective Hamiltonian depends on the phasing, and then the obtaining of qualitative properties of this Hamiltonian with the help of observations via numerical simulations. The third chapter presents results of homogenization in an almost periodic environment. First, we study an evolution problem with a stationary Hamiltonian, almost periodical in space. Using almost periodical arguments, we carry out in a second time a new proof of the homogenization result of the second chapter. The Hamiltonian is then periodic in time and almost periodic in space. We also have some open questions, especially in the case where the Hamiltonian is almost periodic in time-space, and in the case of a traffic model where the traffic lights are quite close, with therefore a microscopic model between the lights.