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

Published online by Cambridge University Press:  01 March 2012

Cyril Imbert ,
Régis Monneau  and
Hasnaa Zidani
Cyril Imbert
Affiliation:
Université Paris-Dauphine, CEREMADE, UMR CNRS 7534, place de Lattre de Tassigny, 75775 Paris Cedex 16, France. imbert@ceremade.dauphine.fr École Normale Supérieure, Département de Mathématiques et Applications, UMR 8553, 45 rue d’Ulm, 75230 Paris Cedex 5, France
Régis Monneau
Affiliation:
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; monneau@cermics.enpc.fr
Hasnaa Zidani
Affiliation:
ENSTA ParisTech & INRIA Saclay (Commands INRIA team), 32 boulevard Victor, 75379 Paris Cedex 15, France; Hasnaa.Zidani@ensta.fr
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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 equationsdiscontinuous Hamiltoniansviscosity solutionsoptimal controltraffic problemsjunctions
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 1 , January 2013 , pp. 129 - 166
Copyright
© EDP Sciences, SMAI, 2012

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References

Y. Achdou, F. Camilli, A. Cutri and N. Tchou, Hamilton-Jacobi equations on networks. Tech. Rep., preprint HAL 00503910 (2010).
Y. Achdou, F. Camilli, A. Cutri and N. Tchou, Hamilton-Jacobi equations on networks, in 18th IFAC World Congress. Milano, Italy (2011).
Bachmann, F. and Vovelle, J., Existence and uniqueness of entropy solution of scalar conservation laws with a flux function involving discontinuous coefficients. Comm. Partial Differential Equations 31 (2006) 371–395. Google Scholar
M. Bardi and I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Systems & Control : Foundations & Applications, Birkhäuser Boston Inc., Boston, MA (1997). With appendices by Maurizio Falcone and Pierpaolo Soravia.
Bardos, C., le Roux, A.Y. and Nédélec, J.-C., First order quasilinear equations with boundary conditions. Comm. Partial Differential Equations 4 (1979) 1017–1034. Google Scholar
Barles, G., Discontinuous viscosity solutions of first-order Hamilton-Jacobi equations : a guided visit. Nonlinear Anal. 20 (1993) 1123–1134. Google Scholar
G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi, Mathématiques & Applications (Berlin) [Mathematics & Applications] 17, Springer-Verlag, Paris (1994).
Bressan, A. and Hong, Y., Optimal control problems on stratified domains. Netw. Heterog. Media 2 (2007) 313–331 (electronic). Google Scholar
Briani, A. and Davini, A., Monge solutions for discontinuous Hamiltonians. ESAIM : COCV 11 (2005) 229–251 (electronic). Google Scholar
Bürger, R. and Karlsen, K.H., Conservation laws with discontinuous flux : a short introduction. J. Engrg. Math. 60 (2008) 241–247. Google Scholar
F. Camilli and D. Schieborn, Viscosity solutions of eikonal equations on topological networks. Preprint.
Chen, X. and Hu, B., Viscosity solutions of discontinuous Hamilton-Jacobi equations. Interfaces and Free Boundaries 10 (2008) 339–359. Google Scholar
Coclite, G.M. and Risebro, N.H., Viscosity solutions of Hamilton-Jacobi equations with discontinuous coefficients. J. Hyperbolic Differ. Equ. 4 (2007) 771–795. Google Scholar
Dal Maso, G. and Frankowska, H., Value function for Bolza problem with discontinuous Lagrangian and Hamilton-Jacobi inequalities. ESAIM : COCV 5 (2000) 369–394. Google Scholar
Engel, K.-J., Fijavž, M. Kramar, Nagel, R. and Sikolya, E., Vertex control of flows in networks. Netw. Heterog. Media 3 (2008) 709–722. Google Scholar
Fathi, A., Théorème KAM faible et théorie de Mather sur les systèmes Lagrangiens. C. R. Acad. Sci. Paris, Sér. I Math. 324 (1997) 1043–1046. Google Scholar
M. Garavello and B. Piccoli, Traffic flow on networks, AIMS Series on Applied Mathematics 1. American Institute of Mathematical Sciences (AIMS), Springfield, MO (2006). Conservation laws models.
Garavello, M. and Piccoli, B., Conservation laws on complex networks. Ann. Inst. Henri. Poincaré, Anal. Non Linéaire 26 (2009) 1925–1951. Google Scholar
Garavello, M. and Soravia, P., Optimality principles and uniqueness for Bellman equations of unbounded control problems with discontinuous running cost. Nonlinear Differential Equations Appl. 11 (2004) 271–298. Google Scholar
Garavello, M. and Soravia, P., Representation formulas for solutions of the HJI equations with discontinuous coefficients and existence of value in differential games. J. Optim. Theory Appl. 130 (2006) 209–229. Google Scholar
Garavello, M., Natalini, R., Piccoli, B., and Terracina, A., Conservation laws with discontinuous flux. Netw. Heterog. Media 2 (2007) 159–179. Google Scholar
Lebacque, J.-P., The Godunov scheme and what it means for first order traffic flow models. Internaional Symposium on Transportation and Traffic Theory 13 (1996) 647–677. Google Scholar
J.-P. Lebacque and M.M. Khoshyaran, Modelling vehicular traffic flow on networks using macroscopic models, in Finite volumes for complex applications II. Hermes Sci. Publ., Paris (1999) 551–558.
J.-P. Lebacque and M.M. Khoshyaran, First order macroscopic traffic flow models : intersection modeling, network modeling, in Transportation and Traffic Theory, Flow, Dynamics and Human Interaction. Elsevier (2005) 365–386.
Lighthill, M.J. and Whitham, G.B., On kinematic waves II. A theory of traffic flow on long crowded roads. Proc. Roy. Soc. A 229 (1955) 317–145. Google Scholar
P.-L. Lions, Generalized solutions of Hamilton-Jacobi equations, Research Notes in Math. 69. Pitman Advanced Publishing Program, Mass Boston (1982).
Lions, P.-L. and Souganidis, P.E., Differential games, optimal control and directional derivatives of viscosity solutions of Bellman’ and Isaacs’ equations. SIAM J. Control Optim. 23 (1985) 566–583. Google Scholar
Newcomb II, R.T. and Su, J., Eikonal equations with discontinuities. Differential Integral Equations 8 (1995) 1947–1960. Google Scholar
Ostrov, D.N., Solutions of Hamilton-Jacobi equations and scalar conservation laws with discontinuous space-time dependence. J. Differential Equations 182 (2002) 51–77. Google Scholar
Richards, P.I., Shock waves on the highway. Operation Research 4 (1956) 42–51. Google Scholar
D. Schieborn, Viscosity Solutions of Hamilton-Jacobi Equations of Eikonal Type on Ramified Spaces. Ph.D. thesis, Eberhard-Karls-Universitat Tubingen (2006).
Seguin, N. and Vovelle, J., Analysis and approximation of a scalar conservation law with a flux function with discontinuous coefficients. Math. Models Methods Appl. Sci. 13 (2003) 221–257. Google Scholar
Siconolfi, A., Metric character of Hamilton-Jacobi equations. Trans. Am. Math. Soc. 355 (2003) 1987–2009 (electronic). Google Scholar
Soravia, P., Discontinuous viscosity solutions to Dirichlet problems for Hamilton-Jacobi equations with convex Hamiltonians. Comm. Partial Differential Equations 18 (1993) 1493–1514. Google Scholar
Soravia, P., Optimality principles and representation formulas for viscosity solutions of Hamilton-Jacobi equations. I. Equations of unbounded and degenerate control problems without uniqueness. Adv. Differential Equations 4 (1999) 275–296. Google Scholar
Soravia, P., Optimality principles and representation formulas for viscosity solutions of Hamilton-Jacobi equations. II. Equations of control problems with state constraints. Differential Integral Equations 12 (1999) 275–293. Google Scholar
Soravia, P., Boundary value problems for Hamilton-Jacobi equations with discontinuous Lagrangian. Indiana Univ. Math. J. 51 (2002) 451–477. Google Scholar
Soravia, P., Uniqueness results for fully nonlinear degenerate elliptic equations with discontinuous coefficients. Commun. Pure Appl. Anal. 5 (2006) 213–240. Google Scholar
Strömberg, T., On viscosity solutions of irregular Hamilton-Jacobi equations. Arch. Math. (Basel) 81 (2003) 678–688. Google Scholar