  Optimal control is a valuable tool for quantum simulation, allowing for the
optimized preparation, manipulation, and measurement of quantum states. Through
the optimization of a time-dependent control parameter, target states can be
prepared to initialize or engineer specific quantum dynamics. In this work, we
focus on the tailoring of a unitary evolution leading to the stroboscopic
stabilization of quantum states of a Bose-Einstein condensate in an optical
lattice. We show how, for states with space and time symmetries, such an
evolution can be derived from the initial state-preparation controls; while for
a general target state we make use of quantum optimal control to directly
generate a stabilizing Floquet operator. Numerical optimizations highlight the
existence of a quantum speed limit for this stabilization process, and our
experimental results demonstrate the efficient stabilization of a broad range
of quantum states in the lattice.
