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

@article{COCV_2013__19_1_129_0, author = {Imbert, Cyril and Monneau, R\'egis and Zidani, Hasnaa}, title = {A {Hamilton-Jacobi} approach to junction problems and application to traffic flows}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {129--166}, publisher = {EDP-Sciences}, volume = {19}, number = {1}, year = {2013}, doi = {10.1051/cocv/2012002}, mrnumber = {3023064}, zbl = {1262.35080}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv/2012002/} }

TY - JOUR AU - Imbert, Cyril AU - Monneau, Régis AU - Zidani, Hasnaa TI - A Hamilton-Jacobi approach to junction problems and application to traffic flows JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2013 SP - 129 EP - 166 VL - 19 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2012002/ DO - 10.1051/cocv/2012002 LA - en ID - COCV_2013__19_1_129_0 ER -

%0 Journal Article %A Imbert, Cyril %A Monneau, Régis %A Zidani, Hasnaa %T A Hamilton-Jacobi approach to junction problems and application to traffic flows %J ESAIM: Control, Optimisation and Calculus of Variations %D 2013 %P 129-166 %V 19 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2012002/ %R 10.1051/cocv/2012002 %G en %F COCV_2013__19_1_129_0

Imbert, Cyril; Monneau, Régis; Zidani, Hasnaa. A Hamilton-Jacobi approach to junction problems and application to traffic flows. ESAIM: Control, Optimisation and Calculus of Variations, Volume 19 (2013) no. 1, pp. 129-166. doi : 10.1051/cocv/2012002. http://www.numdam.org/articles/10.1051/cocv/2012002/

[1] Hamilton-Jacobi equations on networks. Tech. Rep., preprint HAL 00503910 (2010).

, , and ,[2] Hamilton-Jacobi equations on networks, in 18th IFAC World Congress. Milano, Italy (2011).

, , and ,[3] 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. | MR | Zbl

and ,[4] 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. | Zbl

and ,[5] A.Y. le Roux and J.-C. Nédélec, First order quasilinear equations with boundary conditions. Comm. Partial Differential Equations 4 (1979) 1017-1034. | MR | Zbl

,[6] Discontinuous viscosity solutions of first-order Hamilton-Jacobi equations : a guided visit. Nonlinear Anal. 20 (1993) 1123-1134. | MR | Zbl

,[7] Solutions de viscosité des équations de Hamilton-Jacobi, Mathématiques & Applications (Berlin) [Mathematics & Applications] 17, Springer-Verlag, Paris (1994). | MR | Zbl

,[8] Optimal control problems on stratified domains. Netw. Heterog. Media 2 (2007) 313-331 (electronic). | MR | Zbl

and ,[9] Monge solutions for discontinuous Hamiltonians. ESAIM : COCV 11 (2005) 229-251 (electronic). | Numdam | MR | Zbl

and ,[10] Conservation laws with discontinuous flux : a short introduction. J. Engrg. Math. 60 (2008) 241-247. | MR | Zbl

and ,[11] Viscosity solutions of eikonal equations on topological networks. Preprint. | MR | Zbl

and ,[12] Viscosity solutions of discontinuous Hamilton-Jacobi equations. Interfaces and Free Boundaries 10 (2008) 339-359. | MR | Zbl

and ,[13] Viscosity solutions of Hamilton-Jacobi equations with discontinuous coefficients. J. Hyperbolic Differ. Equ. 4 (2007) 771-795. | MR | Zbl

and ,[14] Value function for Bolza problem with discontinuous Lagrangian and Hamilton-Jacobi inequalities. ESAIM : COCV 5 (2000) 369-394. | Numdam | MR | Zbl

and ,[15] Vertex control of flows in networks. Netw. Heterog. Media 3 (2008) 709-722. | MR | Zbl

, , and ,[16] Théorème KAM faible et théorie de Mather sur les systèmes Lagrangiens. C. R. Acad. Sci. Paris, Sér. I Math. 324 (1997) 1043-1046. | MR | Zbl

,[17] Traffic flow on networks, AIMS Series on Applied Mathematics 1. American Institute of Mathematical Sciences (AIMS), Springfield, MO (2006). Conservation laws models. | MR | Zbl

and ,[18] Conservation laws on complex networks. Ann. Inst. Henri. Poincaré, Anal. Non Linéaire 26 (2009) 1925-1951. | Numdam | MR | Zbl

and ,[19] Optimality principles and uniqueness for Bellman equations of unbounded control problems with discontinuous running cost. Nonlinear Differential Equations Appl. 11 (2004) 271-298. | MR | Zbl

and ,[20] 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. | MR | Zbl

and ,[21] Conservation laws with discontinuous flux. Netw. Heterog. Media 2 (2007) 159-179. | MR | Zbl

, , , and ,[22] The Godunov scheme and what it means for first order traffic flow models. Internaional Symposium on Transportation and Traffic Theory 13 (1996) 647-677.

,[23] Modelling vehicular traffic flow on networks using macroscopic models, in Finite volumes for complex applications II. Hermes Sci. Publ., Paris (1999) 551-558. | MR | Zbl

and ,[24] First order macroscopic traffic flow models : intersection modeling, network modeling, in Transportation and Traffic Theory, Flow, Dynamics and Human Interaction. Elsevier (2005) 365-386. | Zbl

and ,[25] On kinematic waves II. A theory of traffic flow on long crowded roads. Proc. Roy. Soc. A 229 (1955) 317-145. | MR | Zbl

and ,[26] Generalized solutions of Hamilton-Jacobi equations, Research Notes in Math. 69. Pitman Advanced Publishing Program, Mass Boston (1982). | MR | Zbl

,[27] Differential games, optimal control and directional derivatives of viscosity solutions of Bellman' and Isaacs' equations. SIAM J. Control Optim. 23 (1985) 566-583. | MR | Zbl

and ,[28] Eikonal equations with discontinuities. Differential Integral Equations 8 (1995) 1947-1960. | MR | Zbl

and ,[29] Solutions of Hamilton-Jacobi equations and scalar conservation laws with discontinuous space-time dependence. J. Differential Equations 182 (2002) 51-77. | MR | Zbl

,[30] Shock waves on the highway. Operation Research 4 (1956) 42-51. | MR

,[31] Viscosity Solutions of Hamilton-Jacobi Equations of Eikonal Type on Ramified Spaces. Ph.D. thesis, Eberhard-Karls-Universitat Tubingen (2006).

,[32] Analysis and approximation of a scalar conservation law with a flux function with discontinuous coefficients. Math. Models Methods Appl. Sci. 13 (2003) 221-257. | MR | Zbl

and ,[33] Metric character of Hamilton-Jacobi equations. Trans. Am. Math. Soc. 355 (2003) 1987-2009 (electronic). | MR | Zbl

,[34] Discontinuous viscosity solutions to Dirichlet problems for Hamilton-Jacobi equations with convex Hamiltonians. Comm. Partial Differential Equations 18 (1993) 1493-1514. | MR | Zbl

,[35] 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. | MR | Zbl

,[36] 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. | MR | Zbl

,[37] Boundary value problems for Hamilton-Jacobi equations with discontinuous Lagrangian. Indiana Univ. Math. J. 51 (2002) 451-477. | MR | Zbl

,[38] Uniqueness results for fully nonlinear degenerate elliptic equations with discontinuous coefficients. Commun. Pure Appl. Anal. 5 (2006) 213-240. | MR | Zbl

,[39] On viscosity solutions of irregular Hamilton-Jacobi equations. Arch. Math. (Basel) 81 (2003) 678-688. | MR | Zbl

,*Cited by Sources: *