  In this article, we give a complete characterization of all the unitary
transformations that can be synthesized in a given time for a system of coupled
spin-1/2 in presence of general time varying coupling tensor. Our treatment is
quite general and our results help to characterize the reachable set at all
times for a class of bilinear control systems with time varying drift and
unbounded control amplitude. These results are of fundamental interest in
geometric control theory and have applications to control of coupled spins in
solid state NMR spectroscopy.
