There have been some applications of optimal control theory to boundary value problems for ordinary differential equations. Among previous works, the best lengths of intervals on which the boundary value problem admits a solution are estimated by Pontryagin’s maximum principle. Hence such approaches are local and the presented conditions are actually not across points of resonance as in the Lazer-Leach condition. Here we consider the existence-uniqueness problem in a class of Neumann boundary value problems for second-order ordinary differential equations probably across several points of resonance. By the optimal control theory method and a careful analysis, we obtain some global optimality results about the existence and uniqueness of solutions for boundary value problems.