Abstract. We consider continuous-state and continuous-time control problems where the admissible trajectories of the system are constrained to remain on a network. A notion of viscosity solution of Hamilton-Jacobi equations on the network has been proposed in earlier articles. Here, we propose a simple proof of a comparison principle based on arguments from the theory of optimal control. We also discuss stability of viscosity solutions.Résumé. On considère des problèmes de contrôle optimal pour lesquels l'état est contraintà rester sur un réseau. Une notion de solution de viscosité deséquations de Hamilton-Jacobi associées aété proposée dans des travaux antérieurs. Ici, on propose une preuve simple d'un principe de comparaison. Cette preuve est basée sur des arguments de contrôle optimal. La stabilité des solutions de viscosité est aussiétudiée.1991 Mathematics Subject Classification. 34H05, 49J15.The dates will be set by the publisher.

We consider a family of optimal control problems in the plane with dynamics and running costs possibly discontinuous across an oscillatory interface Γ ε . The oscillations of the interface have small period and amplitude, both of the order of ε, and the interfaces Γ ε tend to a straight line Γ. We study the asymptotic behavior as ε → 0. We prove that the value function tends to the solution of Hamilton-Jacobi equations in the two half-planes limited by Γ, with an effective transmission condition on Γ keeping track of the oscillations of Γ ε .