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
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France, UMA - Unité de Mathématiques Appliquées
5 UMA - Unité de Mathématiques Appliquées
Abstract : 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.
Keywords : Hamilton-Jacobi equations Discontinuous Hamiltonians Viscosity solutions Optimal control Traffic problems Junctions
Type de document :
Article dans une revue
ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2013, 19 (01), pp 129-166. 〈10.1051/cocv/2012002〉
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Cyril Imbert, Régis Monneau, Hasnaa Zidani. A Hamilton-Jacobi approach to junction problems and application to traffic flows. ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2013, 19 (01), pp 129-166. 〈10.1051/cocv/2012002〉. 〈hal-00569010v3〉

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