  Achieving full control of the time-evolution of a many-body quantum system is
currently a major goal in physics. In this work we investigate the different
ways in which the controllability of a quantum system can be influenced by its
complexity, or even its chaotic properties. By using optimal control theory, we
are able to derive the control fields necessary to drive various physical
processes in a spin chain. Then, we study the spectral properties of such
fields and how they relate to different aspects of the system complexity. We
find that the spectral bandwidth of the fields is, quite generally, independent
of the system dimension. Conversely, the spectral complexity of such fields
does increase with the number of particles. Nevertheless, we find that the
regular o chaotic nature of the system does not affect signficantly its
controllability.
