  We study quantum control of the full hyperfine manifold in the
ground-electronic state of alkali atoms based on applied radio frequency and
microwave fields. Such interactions should allow essentially decoherence-free
dynamics and the application of techniques for robust control developed for NMR
spectroscopy. We establish the conditions under which the system is
controllable in the sense that one can generate an arbitrary unitary on the
system. We apply this to the case of $^{133}$Cs with its $d=16$ dimensional
Hilbert space of magnetic sublevels in the $6S_{1/2}$ state, and design control
waveforms that generate an arbitrary target state from an initial fiducial
state. We develop a generalized Wigner function representation for this space
consisting of the direct sum of two irreducible representation of SU(2),
allowing us to visualize these states. The performance of different control
scenarios is evaluated based on the ability to generate high-fidelity operation
in an allotted time with the available resources. We find good operating points
commensurate with modest laboratory requirements.
