Optimal control of perturbed Hamiltonian systems in R-2 is studied. Systems are considered with a control term scaling with the size of a small perturbing noise. The dynamics are shown to converge in a certain sense to a diffusion on a graph. Using the approach developed in [M. I. Freidlin and A. D. Wentzell, Mem. Amer. Math. Soc., 109 (1994), pp. 1-82] and [M. I. Freidlin and A. D. Wentzell, Ann. Probab., 21 (1993), pp. 2215-2245] for random perturbations of Hamiltonian systems, a convergence theorem is discussed. An optimal control theorem is then developed to maximize the expected exit time from a domain. This control is asymptotically robust for small noise. Several examples are provided.