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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 or chaotic nature of the system does not affect significantly its controllability.