  Time-periodic driving fields could endow a system with peculiar topological
and transport features. In this work, we find dynamically controlled
localization transitions and mobility edges in non-Hermitian quasicrystals via
shaking the lattice periodically. The driving force dresses the hopping
amplitudes between lattice sites, yielding alternate transitions between
localized, mobility edge and extended non-Hermitian quasicrystalline phases. We
apply our Floquet engineering approach to five representative models of
non-Hermitian quasicrystals, obtain the conditions of photon-assisted
localization transitions and mobility edges, and find the expressions of
Lyapunov exponents for some models. We further introduce topological winding
numbers of Floquet quasienergies to distinguish non-Hermitian quasicrystalline
phases with different localization nature. Our discovery thus extend the study
of quasicrystals to non-Hermitian Floquet systems, and provide an efficient way
of modulating the topological and transport properties of these unique phases.
