Volume 19, Number 1, January-March 2013
|Page(s)||129 - 166|
|Published online||01 March 2012|
A Hamilton-Jacobi approach to junction problems and application to traffic flows∗
Université Paris-Dauphine, CEREMADE, UMR CNRS 7534,
place de Lattre de Tassigny,
Paris Cedex 16,
2 École Normale Supérieure, Département de Mathématiques et Applications, UMR 8553, 45 rue d’Ulm, 75230 Paris Cedex 5, France
3 Université Paris-Est, École des Ponts ParisTech, CERMICS, 6 et 8 avenue Blaise Pascal, Cité Descartes Champs-sur-Marne, 77455 Marne-La-Vallée Cedex 2, France
4 ENSTA ParisTech & INRIA Saclay (Commands INRIA team), 32 boulevard Victor, 75379 Paris Cedex 15, France
Received: 12 July 2011
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.
Mathematics Subject Classification: 35F21 / 35D40 / 35Q93 / 35R05 / 35B51
Key words: Hamilton-Jacobi equations / discontinuous Hamiltonians / viscosity solutions / optimal control / traffic problems / junctions
© EDP Sciences, SMAI, 2012
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