In this talk, I will present a representation formula for traffic flow modeling on networks, using Hamilton-Jacobi equations (HJ). It is well-known that if the Hamiltonian does not depend on the space variable, then we can compute explicitly the solution thanks to the so-called Lax-Hopf formula. This formula has been shown to provide efficient tools for traffic flow estimation and control.
In this talk, I will talk about how one can extend such a representation formula in the case where the Hamiltonian does depend on the space variable. Indeed, this means that we could compute explicitly some meaningful macroscopic traffic flow variables even on a network encompassing junctions and traffic discontinuities (bottlenecks) or sources (off and on-ramps).
It is a joint work with Jean-Patrick Lebacque (IFSTTAR).