Ci-dessous vous trouverez le programme des journées ANR COSS. Elle auront lieu dans la salle 1016, batiment Sophie Germain sur le campus Paris Cité.

### Programme du Jeudi 16 mars 2023

**10h30 - 12h(30) ** |
**Présentation sur les aspects administratifs.** |

**12h30 - 14h ** |
**Repas.** |

**14h - 14h45 ** |
** Some ideas to solve perilous problems for viscosity solutions: equations with discontinuities,** Guy Barles, Slides **Résumé** : * 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 **Résumé** : * 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 ** | **Pauses et discussions.** |

**16h15 - 17h ** |
** Viscosity solutions for Hamilton-Jacobi equations in some metric spaces,** Hasnaa Zidani, Slides **Résumé** : * 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 **Résumé** : * 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. * |

### Programme du Vendredi 17 mars 2023

**9h30 - 10h15 ** |
** Homogenization of Hamilton–Jacobi equations on networks,** Antonio Siconolfi, Slides **Résumé** : * 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 ** | **Pauses et 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 **Résumé** : * 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 ** |
**Repas.** |

**14h - 14h45 ** |
** First order Mean Field Games on networks,** Claudio Marchi, Slides
**Résumé** : * 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 **Résumé** : * 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. * |