  Strong synthetic magnetic fields have been successfully implemented in
periodically driven optical lattices. However, the interplay of the driving and
interactions introduces detrimental heating, and for this reason it is still
challenging to reach a fractional quantum Hall state in cold-atom setup. By
performing a numerical study, we investigate stability of a bosonic Laughlin
state in a small atomic sample exposed to driving. We identify an optimal
regime of microscopic parameters, in particular interaction strength $U$ and
the driving frequency $\omega$, such that the stroboscopic dynamics supports
the basic $\nu = 1/2$ Laughlin state. Moreover, we explore slow ramping of a
driving term and show that the considered protocol allows for the preparation
of the Laughlin state on experimentally realistic time scales.
