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The goal of this paper is to derive rigorously macroscopic traffic flow models from microscopic models. More precisely, for the microscopic models, we consider follow-the-leader type models with different types of drivers and vehicles which are distributed randomly on the road. After a rescaling, we show that the cumulative distribution function converge to the solution of a macroscopic model. We also make the link between this macroscopic model and the so-called LWR model.

In this paper, we study a nonlocal approximation of the time-dependent (local) Eikonal equation with Dirichlet-type boundary conditions, where the kernel in the nonlocal problem is properly scaled. Based on the theory of viscosity solutions, we prove existence and uniqueness of the viscosity solutions of both the local and nonlocal problems, as well as regularity properties of these solutions in time and space. We then derive error bounds between the solution to the nonlocal problem and that of the local one, both in continuous time and forward Euler time discretization. We then turn to studying continuum limits of nonlocal problems defined on random weighted graphs with $n$ vertices. In particular, we establish that if the kernel scale parameter decreases at an appropriate rate as $n$ grows then, almost surely, the solution of the problem on graphs converges uniformly to the viscosity solution of the local problem as the time step vanishes and the number vertices $n$ grows large.

International audience; 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 one. Finally, we provide some numerical illustrations.

In this paper, we propose junction conditions for discontinuities due to local perturbation, diverging, merging, and multi-in-multi-out junctions. Traffic flows on junctions can be described by a system of coupled Hamilton-Jacobi equations. At their connection points, it is necessary to propose appropriate junction conditions to close the system. Then, we provide an effective numerical method to compute approximate solutions to these Hamilton-Jacobi equations on junctions. The numerical boundary conditions to close the Hamilton-Jacobi system are also proposed. Numerical tests demonstrate the effectiveness of both the proposed junction conditions and the numerical method.

In this paper we consider an Hamilton-Jacobi equation on a moving in time domain. The boundary is described by a \begin{document}$C^{1}$\end{document} function. We show how we derive this equation from the work of [ 26 ]. We only prove a comparison principle since the proof of other theoritical results can be found in [ 20 ]. At the end of the paper, we consider a short homogenization result in order to reinforce the traffic flow interpretation of the equation.

We present a semi-Lagrangian scheme for the approximation of a class of Hamilton--Jacobi--Bellman (HJB) equations on networks. The scheme is explicit, consistent, and stable for large time steps. W...

In the present paper we prove an homogenisation result for a locally perturbed transport stochastic equation. The model is similar to the stochastic Burgers' equation and it is inspired by the LWR model. Therefore, the interest in studying this equation comes from it's application for traffic flow modelling. In the first part of paper we study the inhomogeneous equation. More precisely we give an existence and uniqueness result for the solution. The technical difficulties of this part come from the presence of the function \begin{document}$ \varphi $\end{document} under assumptions coherent for the model, which is giving the inhomogeneity with respect to the space variable, not present in the classical results. The second part of the paper is the homogenisation result in space.

The goal of this paper is to derive rigorously a macroscopic traffic flow model, for a simple bifurcation, from a microscopic model. At the microscopic scale, we consider a first order model of the form “follow the leader” i.e. the velocity of each vehicle depends on the distance to the vehicle in front of it. We consider the case of a very simple bifurcation in which one road separates into two and one vehicle over two goes to the right and the other goes to the left. At the bifurcation, we then have to add a phase of transition because the vehicle in front will change. Moreover, we assume that the velocity on each road can be different. At the macroscopic scale, we obtain an explicit Hamilton-Jacobi equation on each road and a junction condition (in the sense of [1] ) located at the bifurcation. From this case of a simple bifurcation, we then extend to more general scenarios. For instance, the case of a different distribution of the vehicles at the bifurcation or even to consider more than two outgoing roads. For these extensions we only present the results and explain how to adapt the proofs from the case of a simple bifurcation. Up to our knowledge, this is the first rigorous result for micro-macro passage on a junction.

In this paper, we focus on deriving traffic flow macroscopic models from microscopic models containing a local perturbation such as a traffic light. At the microscopic scale, we consider a first order model of the form "follow the leader" i.e. the velocity of each vehicle depends on the distance to the vehicle in front of it. We consider a local perturbation located at the origin that slows down the vehicles. At the macroscopic scale, we obtain an explicit Hamilton-Jacobi equation left and right of the origin and a junction condition at the origin (in the sense of [ 25 ]) which keeps the memory of the local perturbation. As it turns out, the macroscopic model is equivalent to a LWR model, with a flux limiting condition at the junction. Finally, we also present qualitative properties concerning the flux limiter at the junction.

International audience; The goal of this paper is to derive a traffic flow macroscopic model from a second order microscopic model with a local perturbation. At the microscopic scale, we consider a Bando model of the type following the leader, i.e the acceleration of each vehicle depends on the distance of the vehicle in front of it. We consider also a local perturbation like an accident at the roadside that slows down the vehicles. After rescaling, we prove that the " cumulative distribution functions " of the vehicles converges towards the solution of a macroscopic ho-mogenized Hamilton-Jacobi equation with a flux limiting condition at junction which can be seen as a LWR (Lighthill-Whitham-Richards) model.

International audience; In this article, we study the existence and the uniqueness of traveling waves for a discrete reaction-diffusion equation with bistable non-linearity, namely a generalization of the fully overdamped Frenkel-Kontorova model. This model consists in a system of ODE's which describes the dynamics of crystal defects in a lattice solids. Under very poor assumptions, we prove the existence of a traveling wave solution and the uniqueness of the velocity of propagation of this traveling wave. The question of the uniqueness of the profile is also studied by proving Strong Maximum Principle or some weak asymptotics on the profile at infinity.

International audience; The goal of this paper is to derive a traffic flow macroscopic model from a microscopic model with a transition function. At the microscopic scale, we consider a first order model of the form “follow the leader” i.e. the velocity of each vehicle depends on the distance to the vehicle in front of it. We consider two different velocities and a transition zone. The transition zone represents a local perturbation operated by a Lipschitz function. After rescaling, we prove that the “cumulative distribution function” of the vehicles converges towards the solution of a macroscopic homogenized Hamilton-Jacobi equation with a flux limiting condition at junction which can be seen as a LWR model.

In this paper, we study the motion of spirals by mean curvature type motion in the (two dimensional) plane. Our motivation comes from dislocation dynamics; in this context, spirals appear when a screw dislocation line reaches the surface of a crystal. The first main result of this paper is a comparison principle for the corresponding parabolic quasi-linear equation. As far as motion of spirals are concerned, the novelty and originality of our setting and results come from the fact that, first, the singularity generated by the attached end point of spirals is taken into account for the first time, and second, spirals are studied in the whole space. Our second main result states that the Cauchy problem is well-posed in the class of sub-linear weak (viscosity) solutions. We also explain how to get the existence of smooth solutions when initial data satisfy an additional compatibility condition., Comment: This new version contains new results: we prove that the weak (viscosity) solutions of the Cauchy problem are in fact smooth. This is a consequence of some gradient estimates in time and space

International audience; In the present paper, we consider nonlinear optimal control problems with constraints on the state of the system. We are interested in the characterization of the value function without any controllability assumption. In the unconstrained case, it is possible to derive a characterization of the value function by means of a Hamilton-Jacobi-Bellman (HJB) equation. This equation expresses the behavior of the value function along the trajectories arriving or starting from any position $x$. In the constrained case, when no controllability assumption is made, the HJB equation may have several solutions. Our first result aims to give the precise information that should be added to the HJB equation in order to obtain a characterization of the value function. This result is very general and holds even when the dynamics is not continuous and the state constraints set is not smooth. On the other hand we study also some stability results for relaxed or penalized control problems.

International audience; In this paper, we consider the fully overdamped Frenkel-Kontorova model. This is an infinite system of coupled first order ODEs. Each ODE represents the microscopic evolution of one particle interacting with its neighbors and submitted to a fixed periodic potential. After a proper rescaling, a macroscopic model describing the evolution of densities of particles is obtained. We get this homogenization result for a general class of Frenkel-Kontorova models. The proof is based on the construction of suitable hull functions in the framework of viscosity solutions.

International audience; In this paper, we propose a segmentation method based on the generalized fast marching method (GFMM) developed by Carlini et al. (submitted). The classical fast marching method (FMM) is a very efficient method for front evolution problems with normal velocity (see also Epstein and Gage, The curve shortening flow. In: Chorin, A., Majda, A. (eds.) Wave Motion: Theory, Modelling and Computation, 1997) of constant sign. The GFMM is an extension of the FMM and removes this sign constraint by authorizing time-dependent velocity with no restriction on the sign. In our modelling, the velocity is borrowed from the Chan-Vese model for segmentation (Chan and Vese, IEEE Trans Image Process 10(2):266-277, 2001). The algorithm is presented and analyzed and some numerical experiments are given, showing in particular that the constraints in the initialization stage can be weakened and that the GFMM offers a powerful and computationally efficient algorithm.

International audience; In this paper we prove the convergence at a large scale of a non-local first order equation to an anisotropic mean curvature motion. This first order equation is an eikonal-type equation with a velocity depending in a non-local way on the solution itself, that arises in the theory of dislocations dynamics. We show that if an anisotropic mean curvature motion is approximated by this type of equations then it is always of variational type, whereas the converse is true only in dimension two.

International audience; The goal of this paper is to study a singular perturbation problem in the framework of optimal control on multi-domains. We consider an optimal control problem in which the controlled system contains a fast and a slow variables. This problem is reformulated as an Hamilton-Jacobi-Bellman (HJB) equation. The main difficulty comes from the fact that the fast variable lives in a multi-domain. The geometric singularity of the multi-domains leads to the discontinuity of the Hamiltonian. Under a controllability assumption on the fast variables, the limit equation (as the velocity of the fast variable goes to infinity) is obtained via a PDE approache and by means of the tools of the control theory.

International audience; In this paper, we study the existence and uniqueness of traveling wave solution for the accelerated Frenkel-Kontorova model. This model consists in a system of ODE that describes the motion particles in interaction. The most important applications we have in mind is the motion of crystal defects called dislocations. For this model, we prove the existence of traveling wave solutions under very weak assumptions. The uniqueness of the velocity is also studied as well as the uniqueness of the profile which used different types of strong maximum principle. As far as we know, this is the first result concerning traveling waves for accelerated, spatially discrete system.

In this paper, we prove the existence and uniqueness of a "steady" spiral moving with forced mean curvature motion. This spiral has a stationary shape and rotates with constant angular velocity. Under appropriate conditions on the initial data, we also show the long time convergence (up to some subsequence in time) of the solution of the Cauchy problem to the steady state. This result is based on a new Liouville result which is of independent interest., Comment: 46 pages

International audience; The present paper studies an optimal control problem governed by measure driven differential systems and in presence of state constraints. The first result shows that using the graph completion of the measure, the optimal solutions can be obtained by solving a reparametrized control problem of absolutely continuous trajectories but with time-dependent state-constraints. The second result shows that it is possible to characterize the epigraph of the reparametrized value function by a Hamilton-Jacobi equation without assuming any controllability assumption.

We consider systems of ODEs that describe the dynamics of particles. Each particle satisfies a Newton law (including the acceleration term) where the force is created by the interactions with the other particles and with a periodic potential. The presence of a damping term allows the system to be monotone. Our study takes into account the fact that the particles can be different. After a proper hyperbolic rescaling, we show that the solutions to this system of ODEs converge to the solution of a macroscopic homogenized Hamilton-Jacobi equation., Comment: 37 pages, to appear in TAMS

International audience; This paper deals with some optimal control problems governed by ordinary differential equations with Radon measures as data and with pointwise state constraints. We study the properties of the value function and obtain its characterization by means of an auxiliary control problem of absolutely continuous trajectories. For this, we use some known techniques of reparametrization and graph completion. We are also interested in the characterization of the value function as the unique constrained viscosity solution of a Hamilton-Jacobi equation with measurable time dependant Hamiltonians.

International audience; In this paper, we consider a Generalized Fast Marching Method (GFMM) as a numerical method to compute dislocation dynamics. The dynamics of a dislocation hyper-surface in $\mathbb R^N$ (with $N=2$ for physical applications) is given by its normal velocity which is a non-local function of the whole shape of the hyper-surface itself. For this dynamics, we show a convergence result of the GFMM as the mesh size goes to zero. We also provide some numerical simulations in dimension $N=2$.

International audience; We consider a target problem for a nonlinear system under state constraints. We give a new continuous level-set approach for characterizing the optimal times and the backward-reachability sets. This approach leads to a characterization via a Hamilton-Jacobi equation, without assuming any controllability assumption. We also treat the case of time-dependent state constraints, as well as a target problem for a two-player game with state constraints. Our method gives a good framework for numerical approximations, and some numerical illustrations are included in the paper.

International audience; This paper is concerned with the homogenization of some particle systems with two-body interactions in dimension one and of dislocation dynamics in higher dimensions. The dynamics of our particle systems are described by some ODEs. We prove that the rescaled ``cumulative distribution function'' of the particles converges towards the solution of a Hamilton-Jacobi equation. In the case when the interactions between particles have a slow decay at infinity as $1/x$, we show that this Hamilton-Jacobi equation contains an extra diffusion term which is a half Laplacian. We get the same result in the particular case where the repulsive interactions are exactly $1/x$, which creates some additional difficulties at short distances. We also study a higher dimensional generalisation of these particle systems which is particularly meaningful to describe the dynamics of dislocations lines. One main result of this paper is the discovery of a satisfactory mathematical formulation of this dynamics, namely a Slepcev formulation. We show in particular that the system of ODEs for particle systems can be naturally imbedded in this Slepcev formulation. Finally, with this formulation in hand, we get homogenization results which contain the particular case of particle systems.

International audience; We prove existence of minimizing movements for the dislocation dynamics evolution law of a propagating front, in which the normal velocity of the front is the sum of a non-local term and a mean curvature term. We prove that any such minimizing movement is a weak solution of this evolution law, in a sense related to viscosity solutions of the corresponding level-set equation. We also prove the consistency of this approach, by showing that any minimizing movement coincides with the smooth evolution as long as the latter exists. In relation with this, we finally prove short time existence and uniqueness of a smooth front evolving according to our law, provided the initial shape is smooth enough.

We present a new fast-marching algorithm for an eikonal equation with a velocity-changing sign. This first order equation models a front propagation in the normal direction. The algorithm is an extension of the fast-marching method in two respects. The first is that the new scheme can deal with a time-dependent velocity, and the second is that there is no restriction on its change in sign. We analyze the properties of the algorithm, and we prove its convergence in the class of discontinuous viscosity solutions. Finally, we present some numerical simulations of fronts propagating in $\mathbb{R}^2$.

International audience; In this paper, we study a non-local coupled system that arises in the theory of dislocations densities dynamics. Within the framework of viscosity solutions, we prove a long time existence and uniqueness result for the solution of this model. We also propose a convergent numerical scheme and we prove a Crandall-Lions type error estimate between the continuous solution and the numerical one. As far as we know, this is the first error estimate of Crandall-Lions type for Hamilton-Jacobi systems. We also provide some numerical simulations.

In this article, we present briefly the mathematical study of the dynamics of line defects called dislocations, in crystals. The mathematical model is an eikonal equation describing the motion of the dislocation line with a velocity which is a non-local function of the whole shape of the dislocation. We present some partial existence and uniqueness results. Finally we also show that the self-dynamics of a dislocation line at large scale is asymptotically described by an anisotropic mean curvature motion.

Dislocations are defects present in crystals that are one the main explanation of the plastic behaviour of metals. We are interested in the collective behaviour of such defects and present here an homogenization result in its simplest form.

This thesis deals with the modeling of local and non local Hamilton-Jacobi equations, and their applications to traffic flow. We are concerned with periodic and stochastic Hamilton-Jacobi equations given at two different scales: microscopic and macroscopic. In fact, at the microscopic scale, we describe the dynamic of each vehicle individually and at the macroscopic scale, we describe the traffic in terms of density. From a modeling point of view, the periodic setting means that all the vehicles are identical. This assumption allows to get very interesting results and to justify macroscopic models but it is not very realistic. In contrast to the periodic setting, we also investigate the stochastic setting in which the type of drivers are randomly distributed. In this work we use the theory of viscosity solution, and we derive macroscopic models from microscopic models. We present the specified periodic and stochastic homogenization of first order evolutive Hamilton-Jacobi equations posed on a junction. The first part of this work contains a periodic homogenization of a Hamilton-Jacobi equation posed on junction. In the second part, we prove a specified stochastic homogenization of first order evolutive Hamilton-Jacobi equations on a very simple junction, i.e the real line with a junction at the origin. The main difficulty and novelty of this work comes from the fact that the hamiltonian is not stationary ergodic. Finally, in the last part, we propose a non-local Hamilton-Jacobi model for traffic flow and we prove the existence and uniqueness of the solution of this model. We also propose a numerical scheme and we prove an error estimate between the continuous solution of this problem and the numerical one., 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 chaque véhicule individuellement alors qu'à l'échelle macroscopique, on décrit le trafic en termes de densité. D'un point de vue modélisation, le cadre périodique signifie que tous les véhicules sont identiques. Cette hypothèse permet d'obtenir des résultats très intéressants et de justifier des modèles macroscopiques mais elle n'est pas très réaliste. Contrairement au cadre périodique, nous étudions également le cadre stochastique dans lequel le type de conducteurs est reparti de façon aléatoire. Dans ce travail, nous utilisons la théorie des solutions de viscosité et nous dérivons des modèles macroscopiques à partir de problèmes microscopiques. Nous présentons des résultats d'homogénéisation précisée des équations d'Hamilton-Jacobi périodiques et stochastiques, de premier ordre posées sur une jonction. La première partie de ce travail contient un résultat d'homogénéisation périodique d'une équation d'Hamilton-Jacobi posée sur une jonction. Dans la deuxième partie, nous démontrons un résultat d'homogénéisation stochastique précisée des équations de Hamilton-Jacobi posées sur la ligne réelle avec une jonction à l'origine. La difficulté principale et la nouveauté de ce travail vient du fait que l'hamiltonien n'est pas stationnaire ergodique. Enfin, dans la dernière partie, nous proposons un modèle du type Hamilton-Jacobi non local pour le trafic routier et nous prouvons l'existence et l'unicité de la solution de ce modèle. Nous proposons également un schéma numérique et nous prouvons une estimation d’erreur entre la solution continue de ce problème et la solution numérique.