In this paper, we establish a rigorous connection between a microscopic and a macroscopic traffic model on a convergent junction. At the microscopic scale, we assume that each driver satisfies the "follow the leader" model before the junction point and, a priori, knows its turn to enter the junction. At the macroscopic scale, we obtain a Hamilton-J...
The transition from regular to apparently chaotic motions is often observed in nonlinear flows. The purpose of this article is to describe a deterministic mechanism by which several smaller scales (or higher frequencies) and new phases can arise suddenly in a nonlinear flow under the impact of forcing terms. This phenomenon is illustrated in the co...
In this thesis we propose some variational methods for the mathematical and numerical analysis of a class of HJ equations. Thanks to the metric character of these equations, the set of subsolution corresponds to the set of 1-Lipschitz functions with respect to the Finsler metric associated to the Hamiltonian. Equivalently, it corresponds to the set...
Cette thèse porte sur la modélisation des équations d'Hamilton-Jacobi, locales et non locales, et leurs applications en trafic routier. On considère dans ce travail des équations d'Hamilton-Jacobi périodiques et stochastiques données à deux échelles différentes : microscopique et macroscopique. À l'échelle microscopique, on décrit la vitesse de cha...
We study an equation structured by age and a phenotypic trait describing the growth process of a population subject to aging, competition between individuals, and mutations. This leads to a renewal equation which occurs in many evolutionary biology problems. We aim to describe precisely the asymp-totic behavior of the solution, to infer properties ...
This work is about some random interface growth models whose microscopic evolution is typically represented by a Markov chain. One of the main purposes is to show the hydrodynamic limit i.e the convergence of the rescaled interface to a deterministic macroscopic interface whose evolution is ruled by a Hamilton-Jacobi equation. Then, we are interest...
Published in Automation and Remote Control
Time-optimal differential games with a lifeline are considered. In such games, there are two sets of interest: the first player tries to guide the system into a target set as soon as possible, while the second player counteracts him and wins if the system reaches another set (called the lifeline). A numerical method for solving time-optimal games w...
In this work we propose a non-local Hamilton-Jacobi model for traffic flow and we prove the existence and uniqueness of the solution of this model. This model is justified as the limit of a rescaled microscopic model. We also propose a numerical scheme and we prove an estimate error between the continuous solution of this problem and the numerical ...
Published in Acta Applicandae Mathematicae
We study a mathematical model describing the growth process of a population structured by age and a phenotypical trait, subject to aging, competition between individuals and rare mutations. Our goals are to describe the asymptotic behavior of the solution to a renewal type equation, and then to derive properties that illustrate the adaptive dynamic...
We study the large-time behavior of bounded from below solutions of parabolic viscous Hamilton-Jacobi Equations in the whole space $\mathbb{R}^N$ in the case of superquadratic Hamiltonians. Existence and uniqueness of such solutions are shown in a very general framework, namely when the source term and the initial data are only bounded from below w...
