## Section: New Results

### Traffic Modeling and Control

Participants : M. Goursat, J.P. Quadrat.

The thesis of N. Farhi dedicated to maxplus modeling of microscopic traffic has been defended in June 2008 where a difficult result has been obtained on the explicit computation of an approximation of the fundamental diagram given by a generalized additive eigenvalue for two roads with a junction. This result gives a good insight to what happens on general networks of roads. In this diagram appears phases which have nice traffic interpretation. This year was dedicated to improve the proof and to publish the results in traffic journal and more mathematical oriented journal.

During this redaction and reviewing process appeared a possible extension of the main equation which is finite dimensional to the infinite dimensional case by passing to the limit on the size of the cells on the road. The eigenvalue of the HJB equation obtained in such a way can be linked with the very classical Lightill Whitam equation and its eigenvalue can be computed explicitly in the case of a circular roads. We are studying now what happens when an intersection appears. It could be an HJB equation on a manifold with a singularity for which it is possible to compute explicitly the eigenvalue. This idea will be explored in the next months.