A Perturbation Problem Involving Singular Perturbations of Domains for Hamilton-Jacobi Equations

摘要

We investigate a singular perturbation for Hamilton-Jacobi equations in anopen subset of two dimensional Euclidean space, where the set is determinedthrough a Hamiltonian function and the Hamilton-Jacobi equations are thedynamic programming equations for optimal control of the Hamiltonian flow ofthe Hamiltonian. We establish the convergence of solutions of theHamilton-Jacobi equations and identify the limit of the solutions as solutionsof systems of ordinary differential equations on a graph. The perturbation issingular in the sense that the domain degenerates to the graph in the limitprocess. Our result can be seen as a perturbation analysis, in the viewpoint ofoptimal control, of the Hamiltonian flow.