A Hamiltonian system associated with an infinite horizon optimal control problem has a local stable manifold under appropriate assumptions. Hence, characteristics, candidates of solutions to the optimal control problem, pass through the local stable manifold of the corresponding Hamiltonian system. This paper employs a graph of the local stable manifold as a terminal condition characteristic satisfies, which leads to the two-point boundary value problem with an iterative procedure for computing a point of the local stable manifold. It contributes to forming a bundle of characteristics around a neighborhood of given a reference characteristic with no conjugate points.