Hamilton-Jacobi equations on networks

We consider continuous-state and continuous-time control problem where the admissible trajectories of the system are constrained to remain on a network. Under suitable assumptions, we prove that the value function is continuous. We define a notion of viscosity solution of Hamilton-Jacobi equations on the network for which we prove a comparison principle. The value function is thus the unique viscosity solution of the Hamilton-Jacobi equation on the network.

[1]  H. Frankowska.  Equations d'Hamilton-Jacobi contingentes , 1987 .

[2]  P. Lions,et al.  Hamilton-Jacobi equations with state constraints , 1990 .

[3]  Dirk Schieborn.  Viscosity solutions of Hamilton-Jacobi equations of eikonal type on ramified spaces , 2006 .

[4]  On the Bellman equations with varying control , 1996, Bulletin of the Australian Mathematical Society.

[5]  Eszter Sikolya,et al.  Vertex control of flows in networks , 2008, Networks Heterog. Media.

[6]  H. Soner.  Optimal control with state-space constraint I , 1986 .

[7]  Rufus Isaacs,et al.  Differential Games , 1965 .

[8]  Hasnaa Zidani,et al.  A Hamilton-Jacobi approach to junction problems and application to traffic flows , 2011, 1107.3250.

[9]  Degenerate Eikonal equations with discontinuous refraction index , 2006 .

[10]  H. Ishii,et al.  A New Formulation of State Constraint Problems for First-Order PDEs , 1996 .

[11]  H. Frankowska.  Hamilton-Jacobi equations: Viscosity solutions and generalized gradients , 1989 .

[12]  H. Frankowska,et al.  Semicontinuous Solutions of Hamilton–Jacobi–Bellman Equations with Degenerate State Constraints , 2000 .

[13]  C. M. Elliott,et al.  Uniqueness and error analysis for Hamilton-Jacobi equations with discontinuities , 2004 .

[14]  P. Varaiya,et al.  Differential games , 1971 .

[15]  M. Bardi,et al.  Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations , 1997 .

[16]  S. Nicaise,et al.  Elliptic operators on elementary ramified spaces , 1988 .

[17]  Y. V. Pokornyi,et al.  Differential Equations on Networks (Geometric Graphs) , 2004 .

[18]  Alberto Bressan,et al.  Optimal control problems on stratified domains , 2007, Networks Heterog. Media.

[19]  Yves Achdou,et al.  Hamilton–Jacobi equations constrained on networks , 2013 .

[20]  A. Tourin.  A comparison theorem for a piecewise Lipschitz continuous Hamiltonian and application to Shape-from-Shading problems , 1992 .

[21]  P. Souganidis,et al.  Differential Games, Optimal Control and Directional Derivatives of Viscosity Solutions of Bellman’s and Isaacs’ Equations , 1985 .

[22]  Viscosity solutions of Eikonal equations on topological networks , 2011, 1103.4041.