  We study the ability to implement unitary maps on states of the $I=9/2$
nuclear spin in \textsuperscript{87}Sr, a $d=10$ dimensional (qudecimal)
Hilbert space, using quantum optimal control. Through a combination of nuclear
spin-resonance and a tensor AC-Stark shift, by solely modulating the phase of a
radio-frequency magnetic field, the system is quantum controllable. Alkaline
earth atoms, such as \textsuperscript{87}Sr, have a very favorable
figure-of-merit for such control due to narrow intercombination lines and the
large hyperfine splitting in the excited states. We numerically study the
quantum speed-limit, optimal parameters, and the fidelity of arbitrary state
preparation and full SU(10) maps, including the presence of decoherence due to
optical pumping induced by the light-shifting laser. We also study the use of
robust control to mitigate some dephasing due to inhomogeneities in the light
shift. We find that with an rf-Rabi frequency of $\Omega_\text{rf}$ and 0.5\%
inhomogeneity in the the light shift we can prepare an arbitrary Haar-random
state in a time $T={4.5}\pi/\Omega_\text{rf}$ with average fidelity $\langle
\mathcal{F}_\psi \rangle =0.9992$, and an arbitrary Haar-random SU(10) map in a
time $T=24\pi/\Omega_\text{rf}$ with average fidelity $\langle \mathcal{F}_U
\rangle = 0.9923$.
