A perturbation problem involving singular perturbations of domains for Hamilton-Jacobi equations

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.