El Khatib, Nader Forcadel, Nicolas Zaydan, Mamdouh

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...

Cheverry, Christophe Farhat, Shahnaz

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...

Ennaji, Hamza

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...

Fayad, Rim

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...

Nordmann, Samuel Perthame, Benoît

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 ...

Lerouvillois, Vincent

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...

Munts, N.V. Kumkov, S.S.
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...

Ciotir, I Fayad, R Forcadel, N Tonnoir, Antoine

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 ...

Nordmann, Samuel Perthame, Benoît Taing, Cécile
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...

Barles, Guy Quaas, Alexander Rodríguez, Andrei

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...