A Hamilton-Jacobi approach to junction problems and application to traffic flows

Cyril Imbert (1, 2) , Régis Monneau (3) , Hasnaa Zidani (4, 5)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 DMA - Département de Mathématiques et Applications - ENS Paris
3 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
4 Commands - Control, Optimization, Models, Methods and Applications for Nonlinear Dynamical Systems
5 UMA - Unité de Mathématiques Appliquées

This paper is concerned with the study of a model case of first order Hamilton-Jacobi equations posed on a ''junction'', that is to say the union of a finite number of half-lines with a unique common point. The main result is a comparison principle. We also prove existence and stability of solutions. The two challenging difficulties are the singular geometry of the domain and the discontinuity of the Hamiltonian. As far as discontinuous Hamiltonians are concerned, these results seem to be new. They are applied to the study of some models arising in traffic flows. The techniques developed in the present article provide new powerful tools for the analysis of such problems.

Mots clés: Hamilton-Jacobi equations; Discontinuous Hamiltonians; Viscosity solutions; Optimal control; Traffic problems; Junctions

Domaines: Equations aux dérivées partielles [math.AP]

Cyril Imbert, Régis Monneau, Hasnaa Zidani. A Hamilton-Jacobi approach to junction problems and application to traffic flows. 2011. ⟨hal-00569010v2⟩