10h - 10h30

Accueil-Café,

10h30 - 11h15

Stratified Problems arising from Homogenization of Hamilton-Jacobi equations, Yves Achdou, Slides

Résumé : We describe several cases in which the homogenization of stationary Hamilton-Jacobi equations leads to stratified problems (the class of stratified problems has been introduced by A. Bressan and Y. Hong and later studied by G. Barles and E. Chasseigne). We first recall results obtained with S. Oudet and N. Tchou, in which the limiting problem in R^d is a stratified problem in which the stratification is of the type R^d \ M_{d-1}, where M_{d-1} is a d-1 dimensional subspace. Next, we consider a class of Hamilton-Jacobi equations in which the Hamiltonian is obtained by perturbing near the origin an otherwise periodic Hamiltonian (collaboration with C. Le Bris). The limiting problem involves the stratification (\R^d\{0}) \cup \{0\}. Finally, we study homogenization for a class of bidimensional stationary Hamilton-Jacobi equations where the Hamiltonian is obtained by perturbing near a half-line a Hamiltonian which does not depend on the fast variable, or depends on the fast variable in a periodic manner. We prove that the limiting problem involves a stratification, made of a submanifold of dimension zero, namely the endpoint of the half-line, a submanifold of dimension one, the open half-line, and the complement of the latter two sets which is a submanifold of dimension two. The limiting problem involves effective Hamiltonians which are associated to the abovementioned three submanifolds and keep track of the perturbation.

11h30 - 12h15

Processus adhérents sur des réseaux et jeux à champ moyen associés, Jules Berry, Slides

Résumé : Dans cet exposé nous présenterons dans un premier temps les principales propriétés des processus de diffusion adhérents (sticky) sur des réseaux. Les processus de diffusions sur les réseaux se comportent comme des processus de diffusion unidimensionnels à l'intérieur de chaque branche et sont réfléchis aléatoirement à l'intérieur d'une autre branche lorsqu'ils atteignent une jonction. Les processus adhérents sont alors caractérisés par le fait que la réflexion se fait de manière lente, ce qui se manifeste par le fait que ces processus passent un temps strictement positif à la jonction. Nous nous intéresserons ensuite aux systèmes de jeux à champ moyen (MFG) du second ordre obtenus lorsque les agents sont représentés par de tels processus. La principale différence avec les MFG basés sur des processus non-adhérents (tels qu'étudiés par Camilli-Marchi et Achdou-Dao-Ley-Tchou) réside dans le fait que, dans notre cas, la densité des agents présente une partie singulière localisée au niveau des jonctions. Travail réalisé en collaboration avec Fabio Camilli et Fausto Colantoni.

12h30 - 14h30

Repas.

14h30 - 15h15

Generalized traveling wave solutions for mean curvature flows with merely L^{\infty} forcing in the plane, Nathaël Alibaud

Résumé : This work is in collaboration with Gawtum Namah. We consider traveling wave solutions of curvature flows in the plane with periodic forcing $R$. The front's profile is the graph of a function $\psi $ satisfying a PDE and typically bounded if $\essinf R>0$. This theory has been extended to generalized waves in [Cesaroni \& Novaga, {\it Comm. Partial Differential Equations,} 2015] for $R \in W^{1,\infty }$ with nonconstant sign. The profile may then be unbounded and the new formulation is variational. Equivalently $\psi$ solves a boundary value problem with infinite Dirichlet condition. We investigate $L^{\infty}$ forcing and obtain profiles with vertical lines. This amounts to consider discontinuous $\psi$ satisfying an infinite Neuman boundary condition complemented by an interface-like condition. We illustrate our results for prototype fibered media, for which we explicitely compute all their generalized waves. We get a necessary and sufficient condition for the profile to be bounded. At the limiting threshold, there are many different profiles with vertical lines of arbitrary heights. We finally show that all these waves are all limits of waves from the $W^{1,\infty}$ theory.

15h30 - 16h15

Approximation of an optimal control problem posed on a network with a perturbed problem in the whole space, Mohamed Camar-Eddine, Slides

Résumé : A classical optimal control problem posed in the whole space~$\mathbb{R}^2$ is perturbed by a singular term of magnitude $\varepsilon^{-1}$ aimed at driving the trajectories closer and closer to a network. We are interested in the link between the limit problem, as $\varepsilon\to0$, and some optimal control problems on networks studied in the literature. This is a joint work with M. Chuberre, M. Haddou and O. Ley..

16h15 - 16h45

Pause-Café.

16h45 - 17h30

Nouvelles représentations pour les jeux différentiels et application à des schémas à base de réseau de neurones pour la fonction valeur, Olivier Bokanowski, Slides

Résumé : Nous étudions des problèmes de contrôle optimal déterministes pour des jeux différentiels à horizon fini, et on cherche à développer un cadre rigoureux pour l'approximation de la fonction valeur par une approche à base de réseaux de neurones. Pour le problème semi-discrétisé en temps (problème à contrôles constants par morceaux), on commence par proposer de de nouvelles représentations de la fonction valeur avec des stratégies non-anticipatives sous forme feedback. On obtient de nouvelles estimations d'erreur entre le problème semi-discrétisé et le problème continu sous certaines hypothèses sur la dynamique (dynamique de type proie-prédateur). On considère alors un schéma d'approximation en espace, à base de réseaux de neurones. On donnera un résultat de convergence du schéma dans un certain sens faible, ainsi que quelques illustrations numériques pour des problèmes d'atteignabilité sous contraintes. Ce travail a été réalisé en collaboration avec Xavier Warin.

9h15 - 10h

Gehring's Lemma for kinetic Fokker-Planck equations, Jessica Guérand,

Résumé : Gehring's Lemma states that a real function satisfying a reverse Hölder inequality on sub-domains has an improved degree of integrability. Motivated by open problems on quasi-conformal mapping, it was initially introduced by Gehring and then adapted and used to study gain integrability on gradient of solutions of elliptic and parabolic equations. Here I will present a joint work with Cyril Imbert and Clément Mouhot in which we establish the result for kinetic Fokker-Planck equations by getting a Gehring lemma for kinetic cylinder sub- domains.

10h - 10h30

Pause-Café,

10h30 - 11h15

Decomposed resolution of multi-agent aggregative optimal control problems, Kang Liu, Slides

Résumé : We are interested in an aggregative deterministic optimal control problem, with discrete state-space, discrete time, and numerous agents. This optimal control problem can be viewed as a special case of an aggregative nonconvex optimization problem that can be solved by the stochastic Frank-Wolfe (SFW) algorithm, whose theoretical convergence results in both the expectation and probability senses are established. In the optimal control scenario, the SFW algorithm amounts to solving at each iteration a series of small-scale optimal control problems, corresponding to each agent. These sub-problems are solved by dynamic programming efficiently. Numerical results are presented, for a model of the charging management of a battery fleet.

11h30 - 12h15

Test function approach to fully nonlinear equations in thin domains with oblique boundary conditions, Ariela Briani, Ariela Briani, Slides

Résumé : In the first joint work with I. Birindelli and H. Ishii [1] we extended to fully nonlinear operators the well known result on thin domains of Hale and Raugel [2]. This result is more general even in the case of the Laplacian. I will describe this result and a work in progress where we consider oblique boundary condition. In this case we find some new phenomena, in particular the limit equations contains "new terms" in the second, first and zero order term which don't have an equivalent in the Neumann case. [1] Isabeau Birindelli, Ariela Briani, Hitoshi Ishii, Test function approach to fully nonlinear equations in thin domains, arXiv:2404.19577, April 2024. [2] Jack K. Hale, Geneviève Raugel, Reaction-diffusion equation on thin domains, J. Math. Pures Appl. (9) 71 (1992), no. 1, 33--95.

12h15 - 14h15

Repas.

14h15 - 15h

Solutions to the Hamilton-Jacobi equation for dynamic optimization problems with discontinuous time dependence, Piernicola Bettiol, Slides

Résumé : We will provide characterizations of the value function as the unique lower semicontinuous solution, appropriately defined, to the Hamilton-Jacobi equation associated with some classes of dynamic optimization problems with discontinuous time dependence. In particular we shall consider state constrained optimal control problems of Bolza type in which the running cost is possibly non Lipschitz continuous w.r.t. the state variable and Calculus of Variations problems in which the functional to minimize comprises an end-point cost function and an integral term involving a nonautonomous Lagrangian. We shall give also some illustrative examples. This talk is based on recent results obtained in collaboration with J. Bernis, C. Mariconda and R. Vinter.

9h15 - 10h

Conditions de couplage d'interface pour les lois de conservation scalaires, Boris Andreianov, Slides

Résumé : La motivation pour cet exposé vient d'un problème de couplage dit "de Kedem-Katchalski" des lois de conservation au noeud d'un réseau (travail en cours de finalisation avec G. Coclite et C. Donadello). Nous proposons de revisiter les représentations établies (les conditions BLN pour les problèmes aux limites, les conditions de germe L1-dissipatif pour les problèmes de "flux discontinu") dans un cadre permettant d'avoir un point de vue commun, et explicitant les hypothèses de modélisation sous-jacentes. La question de monotonie des conditions de couplage formelles (définies au niveau du modèle souhaité de comportement au noeud) apparaîtra comme une question centrale de la théorie. Le solveur de Riemann et le flux de Godunov au noeud seront les principaux outils pour décrire le couplage effectif, résultant d'une procédure de projection connue dans le cas des conditions aux limites de Dirichlet. Afin de définir ledit solveur et ledit flux, nous serons amenés à nous servir de la théorie abstraite des opérateurs m-accrétifs sur $\R^{k}$. Cette approche permet de déterminer les limites d'applicabilité des techniques propres aux lois de conservation, et se s'interroger (en lien avec le récent travail de Cardaliaguet, Forcadel, Girard, Monneau) sur le potentiel de la description Hamilton-Jacobi.

10h - 10h45

Trace du gradient pour HJB, Régis Monneau, Slides

Résumé : Nous considérons des equations de Hamilton-Jacobi avec Hamiltonien strictement convexe sur un domaine (en espace ou en espace-temps). Pour des solutions Lipschitz, nous montrons que si la solution admet un gradient tangentiel en un point du bord, alors elle admet aussi une dérivée normale en ce même point. Une conséquence est l’existence d’une trace forte pour le gradient. Ces résultats s’adaptent au cas où le bord est de codimension plus grande que un (comme un point isolé dans l’espace entier). On montre alors l’existence de dérivées directionnelles, et un contre-exemple lorsque la dépendance du Hamiltonien en la variable d’espace n’est pas assez régulière.

10h45 - 11h

Pause café.

11h - 11h45

Unicité forte pour Hamilton-Jacobi : la méthode des blows-up jumeaux, Cyril Imbert

Résumé : À compléter.

11h45 - 12h30

Localized Inverse Design for Conservation Laws and Hamilton-Jacobi equations, Vincent Perrollaz, Slides

Résumé : À compléter.

12h30 - 13h45

Repas.

13h45 - 14h30

De P_2 à L_2: deux points de vue sur un problème de contrôle et son équation de HJB, Chloé Jimenez, Slides

Résumé : Dans cet exposé, je présenterai un problème de contrôle multi-agent exprimé dans l'espace de Wasserstein étudié dans deux articles de Marigonda, Quincampoix et J., Marigonda, Quincampoix. Je montrerai d'abord comme se problème peut aussi être vu comme un problème de contrôle L²_P. La fonction valeur exprimée dans l'espace de Wasserstein est solution de viscosité d'une équation de Hamilton-Jacobi au sens de Gangbo-Nguyen-Tudorascu. Nous verrons qu'elle est également solution d'une équation de Hamilton-Jacobi dans L²_P, nous construirons l'Hamiltonien de cette équation de façon à ce qu'il soit régulier.

14h30 - 15h15

Equivalence between two notions of viscosity solutions in the Wasserstein space, Averil Prost, Slides

Résumé : We present two notions of viscosity solutions for first-order Hamilton-Jacobi equations, one using test functions that are directionally differentiable, and another one using generalized sub/superdifferentials. In the classical setting of viscosity over IR^d, it is simple to link semidifferentials to the gradients of test functions. We show that a similar equivalence holds in our nonsmooth setting, under appropriate conditions over the Hamiltonian.

15h15 - 15h30

Pause café.

15h30 - 16h15

Numerical approximation of the mean field game problem, Ahmad Zorkot, Slides

Résumé : This talk is devoted to the numerical approximation of mean field games problems. We consider two cases: a first order problem, i.e the diffusion is null, and a second order problem. For the first one, we propose a Lagrange-Galerkin method to approximate the solution of a class of continuity equation, coupled with a semi-Lagrangian discretization of an Hamilton-Jacobi-Bellman equation, in order to obtain an approximation method for a first order Mean Field Games system. We prove a convergence result and we show some numerical simulations. For the second order case, we consider a Newton iterations approach for the continuous mean field game system which result a system of two linear parabolic equation that we solve using two approaches: a semi Lagrangian scheme and a finite difference scheme, we then conduct a comaprative analysis between the mentioned schemes and alternative schemes outlined in existing literature. Joint work with Elisabetta Carlini and Francisco J. Silva.

16h15 - 17h

Approximation de lois de conservation paramétrées, Nicolas Seguin, Slides

Résumé : À compléter.

9h30 - 10h15

Homogenization of Hamilton–Jacobi equations on networks, Antonio Siconolfi, Slides

Abstract : We present an homogenization procedure for time dependent Hamilton–Jacobi equations posed on networks embedded in the Euclidean space RN , and depending on an oscillation parameter ε which becomes infinitesimal. The peculiarity of the construction is that the limit equation is posed in an Euclidean space whose dimension depends on the topological complexity of the network. Approx- imating and limit equations are therefore defined on different spaces, this requires an appropriate notion of convergence for the corresponding solutions. We use closed probability measures defined on an abstract graph underlying the network, and define an equivalent on graph of the so–called Mather α and β functions. The α function plays the role of effective Hamiltonian. The results have been obtained in collaboration with Marco Pozza and Alfonso Sorrentino.

10h15 - 11h

Break and discussions.

11h - 11h45

A multi-population traffic flow model on networks accounting for routing strategies, Paola Goatin, Slides

Résumé : We introduce a macroscopic differential model coupling a conservation laws with a Hamilton-Jacobi equation to respectively model the nonlinear transportation process and the strategic choices of users. Furthermore, the model is adapted to the multi-population case, where every population differs in the level of traffic information about the system. This allows to reproduce well-known phenomena, like Braess’ paradox, and to investigate the impact of navigation systems on the network efficiency.

11h45 - 12h30

Inverse Design for Conservation Laws and Hamilton-Jacobi equations, Vincent Perrollaz, Slides

Abstract : Consider a Conservation Law and a Hamilton-Jacobi equation with a flux/Hamiltonian depending also on the space variable. We characterize first the attainable set of the two equations and, second, the set of initial data evolving at a prescribed time into a prescribed profile. An explicit example then shows the deep differences between the cases of x-independent or x-dependent fluxes/Hamiltonians. This talk is based on joint works with Rinaldo Colombo and Abraham Sylla.

12h30 - 14h

Meal.

14h - 14h45

First order Mean Field Games on networks, Claudio Marchi, Slides

Abstract : We focus our attention on deterministic Mean Field Games with finite horizon in which the states of the players are constrained in a network (in our setting, a network is given by a finite collection of vertices connected by continuous edges) and the cost may change from edge to edge. As in the Lagrangian approach, we introduce a relaxed notion of Mean Field Games equilibria which describe the game in terms of probability measures on trajectories instead of time-dependent probability measures on the network. Our first main result is to establish the existence of such a MFG equilibrium. Afterward, to each MFG equilibrium, can be naturally associated a cost, the corresponding value function and optimal trajectories (chosen by the agents). We prove that optimal trajectories starting at time t=0 are Lipschitz continuous, locally uniformly with respect to the initial position. As a byproduct, we obtain a ``Lipschitz’’ continuity of the MFG equilibrium: its push-forward through the evaluation-function at each time gives rise to a Lipschitz continuous function from the time interval to the space of probability measures on the network. The second main result is to prove that this value function is Lipschitz continuous and solves a Hamilton-Jacobi partial differential equation in the network. This is a joint work with: Y. Achdou (Univ. of Paris), P. Mannucci (Univ. of Padova) and N. Tchou (Univ. of Rennes).

14h45 - 15h30

Microscopic derivation of a traffic flow model with a bifurcation, Pierre Cardaliaguet, Slides

Abstract : The talk will be based on joint works with N. Forcadel, T. Girard and R. Monneau. In a first part I will describe a work with N. Forcadel in which we derive rigorously a macroscopic traffic flow model with a bifurcation or a local perturbation from a microscopic one. The microscopic model is a simple follow-the-leader with random parameters. The random parameters are used as a statistical description of the road taken by a vehicle and its law of motion. The limit model is a deterministic and scalar Hamilton-Jacobi on a network with a flux limiter, the flux-limiter describing how much the bifurcation or the local perturbation slows down the vehicles. The proof of the existence of this flux limiter---the first one in the context of stochastic homogenization---relies on a concentration inequality and on a delicate derivation of a superadditive inequality. In a second part, I will explain the relations between the resulting Hamilton-Jacobi equation and some conservation laws with discontinuous fluxes.

10h30 - 12h(30)

Presentation on the administrative aspects.

12h30 - 14h

Meal.

14h - 14h45

Some ideas to solve perilous problems for viscosity solutions: equations with discontinuities, Guy Barles, Slides

Abstract : Viscosity solutions theory was developed for continuous Hamiltonians and, even if Ishii's definition and the half-relaxed limits method give the impression that the discontinuous framework is at hand, there are major difficulties to handle it: not only the proofs of the comparison results --all based on the famous ``doubling of variables'' method-- completely fail for discontinuous Hamiltonians but it is not difficult to construct examples showing that such comparison results are false in general. This is due to Ishii's definition which is not well-adapted to treat discontinuities and needs to be reinforced on each of them. In this talk, we describe recent developments where the theory is extended for convex Hamiltonians to the case of ``stratified problems'' where the Hamiltonians may have discontinuities on a Whitney stratification. By ``extended'', we mean that, using a correct definition of sub and supersolutions, we recover the key comparison and stability results which are the main pillars of viscosity solutions theory. These results, which generalize those obtained in a similar framework by Bressan and Hong, hold under ``natural assumptions'' and we will justify why we consider them as being natural. These results are part of a book with E. Chasseigne in which we consider also the case of co-dimension 1 discontinuities for which a rather complete set of results exists for convex AND non-convex Hamiltonians, using the notions of flux-limited solutions of Imbert-Monneau or junction viscosity solutions by Lions-Souganidis.

14h45 - 15h30

Conservation laws on a star-shaped network, Carlotta Donadello, Slides

Abstract : Hyperbolic conservation laws defined on oriented graphs are widely used in the modeling of a variety of phenomena such as vehicular and pedestrian traffic, irrigation channels, blood circulation, gas pipelines, structured population dynamics. From the point of view of the mathematical analysis each of these situations demands for a different definition of admissible solution, encoding in particular the node coupling between incoming and outgoing edges which is the most coherent with physical observations. A comprehensive study of the necessary and sufficient properties of the coupling conditions which lead to well-posedness of the corresponding admissible solutions is available in the framework of conservation laws with discontinuous flux, which can be seen as a simple $1-1$ network. A similar theory for conservation laws on star-shaped graph is at its beginning. In particular, the characterization of family of solutions obtained as limits of regularizing approximations, such as vanishing viscosity limits, is still a partially open problem. In this talk we’ll provide a general introduction to the topic, an overview of the most recent results and some explicit examples.

15h30 - 16h15

Break and discussions.

16h15 - 17h

Viscosity solutions for Hamilton-Jacobi equations in some metric spaces, Hasnaa Zidani, Slides

Abstract: In this talk, we will discuss a viscosity notion for solutions of Hamilton-Jacobi equations in some metric spaces. This notion is based on test functions that are directionally differentiable and can be represented as a difference of two semiconvex functions. Under mild assumptions on the Hamiltonian and on the metric space, we can derive the main properties of viscosity theory: the comparison principle and Perron's method.

17h - 17h45

Une discrétisation de type Lagrange-Galerkin pour les jeux à champ moyen du premier ordre, Francisco Silva, Slides

Abstract : Dans cet exposé, basé sur un travail en collaboration avec E. Carlini (Université de Rome I, La Sapienza) et A. Zorkot (Université de Limoges), je présenterai un schéma de discrétisation de type Lagrange-Galerkin pour les jeux à champ moyen du premier ordre (où déterministes). Nous montrons que le schéma admet au moins une solution et nous établissons un résultat de convergence vers une solution du système initial sans imposer de contraintes sur la dimension de l'espace d'états.