HJnet

We propose to launch a research project on Hamilton-Jacobi (HJ) equations on networks, and more generally on heterogeneous structures. This theoretical problem has several potential applications, in particular to traffic flow theory, which is much studied from the point of view of conservation laws, but very little from the viewpoint of HJ equations. Whereas discrete control problems on networks have been much studied in the literature, there is almost nothing written on the case when the running cost and the dynamics vary with respect to time and state. The difficulty lies in the fact that, in a network, the set of admissible controls drastically changes from a point in the interior of an edge, where only one direction is admissible, to a vertex where the admissible directions are given by all the edges connected to it. Therefore, even if the data of the problem are regular, the corresponding Hamiltonian when restricted to the network has a discontinuous structure with respect to the spatial variable. This kind of discontinuity is at the cutting edge of the viscosity theory for HJ equations and most related problems are wide open. We think that there is room for a three years ANR-blanc project on the previously mentioned topic as well as on related problems about HJ equations. There will be four partners: Rennes (IRMAR), Paris 7 (LJJL), École Nationale des Ponts et Chaussées (CERMICS) and Tours (LMPT). We plan to investigate many aspects of the problems, both theoretical and practical. Trying to establish a robust theory of viscosity solutions for HJ equations on networks and heterogeneous structures will be a fundamental task. It will include the study of related and more general problems, such as HJ equations with space discontinuities. Another task will be to study the applications to traffic: we will start with models for traffic on a divergent junction, and gradually complexify the models to include micro-macro approaches, junctions with variable proportions of directions, convergent junctions and priority laws. We plan to compare our results with those obtained using conservation laws. In a third task, we plan to study singular perturbation problems for domains whose thickness tends to zero and see if at the limit, we recover the previously mentioned notion of viscosity solution on networks. We will also consider homogenization problems. In particular, we will study situations when a network becomes denser and denser, to model traffic in cities for example. We will also investigate differential games (leading to nonconvex Hamiltonians) and ask ourselves the same questions about the proper notion of viscosity solutions on networks. We will also consider mean field games on networks, with obvious applications in traffic theory, but also in many other situations. Finally, numerical methods will be proposed and analyzed. The general organization of the project will be classical for a project in mathematics: we will have frequent workshops of two days each in order to organize the research and exchange our ideas. We will also organize a mid-term school+workshop and a final conference in Tours and Rennes in order to invite international experts and to advert our results.