  We investigate a singular perturbation for Hamilton-Jacobi equations in an
open subset of two dimensional Euclidean space, where the set is determined
through a Hamiltonian function and the Hamilton-Jacobi equations are the
dynamic programming equations for optimal control of the Hamiltonian flow of
the Hamiltonian. We establish the convergence of solutions of the
Hamilton-Jacobi equations and identify the limit of the solutions as solutions
of systems of ordinary differential equations on a graph. The perturbation is
singular in the sense that the domain degenerates to the graph in the limit
process. Our result can be seen as a perturbation analysis, in the viewpoint of
optimal control, of the Hamiltonian flow.
