Hamilton-Jacobi Equations on Networks as Limits of Singularly Perturbed Problems in Optimal Control: Dimension Reduction

We consider a family of star-shaped planar domains omega(epsilon), made of N non intersecting semi-infinite strips of thickness epsilon and of a central region whose diameter is proportional to epsilon. As epsilon -> 0, omega(epsilon) tends to a network G made of half-lines sharing an endpoint O. We study infinite horizon optimal control problems in which the state is constrained to remain in . We prove that the value function tends to the solution of a Hamilton-Jacobi equation on G, with an effective transmission condition at O. The effective equation is linked to an optimal control problem.