ANR HJnet  (ANR-12-BS01-0008-01)
Hamilton-Jacobi equations on heterogeneous structures and networks

Slides of the conference "Hamilton-Jacobi Equations : New trends and applications" (final conference of the ANR HJnet), May 30 - June 03, 2016, Rennes

Presentation of the project

The project ANR HJnet is funded by the French National Research Agency (ANR-12-BS01-0008-01). It started in January 2013 and will end in December 2015.

This research project brings together mathematicians of different backgrounds to work on Hamilton-Jacobi 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 continuously 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 space variable. This kind of discontinuity is at the cutting edge of the viscosity theory for HJ equations and most related problems are wide open.

Scientific topics of the project:
  • Theory of viscosity solutions for HJ equations on networks and heterogeneous structures
  • HJ equations with space discontinuities
  • Optimal control on networks
  • Applications to traffic
  • singular perturbation problems
  • Homogenization of HJ equations
  • Differential games
  • Mean field games on networks
  • Numerical methods

Four teams are involved in the project. Members are mainly based in