Optimal population transfers for a quantum system in the limit of large transfer time

We consider an optimal population transfer problem for a finite-dimensional quantum system with an energy-like cost. We show that a way to realize a small control limit is as the limit of large transfer time T. In the process we show that, in the large T limit, the optimal control is a sum of terms with the following structure: each term is an exponential with frequency given by a Bohr frequency of the quantum system times a slow varying envelope, that is a function of t/T. The form of these envelopes can be computed by solving an "averaged" two-point boundary value problem. We demonstrate our results with an example.