  We consider a quantum system periodically driven with a strength which varies
slowly on the scale of the driving period. The analysis is based on a general
formulation of the Floquet theory relying on the extended Hilbert space. It is
shown that the dynamics of the system can be described in terms of a slowly
varying effective Floquet Hamiltonian that captures the long-term evolution, as
well as rapidly oscillating micromotion operators. We obtain a systematic
high-frequency expansion of all these operators. Generalizing the previous
studies, the expanded effective Hamiltonian is now time-dependent and contains
extra terms appearing due to changes in the periodic driving. The same applies
to the micromotion operators which exhibit a slow temporal dependence in
addition to the rapid oscillations. As an illustration, we consider a
quantum-mechanical spin in an oscillating magnetic field with a slowly changing
direction. The effective evolution of the spin is then associated with
non-Abelian geometric phases reflecting the geometry of the extended Floquet
space. The developed formalism is general and also applies to other
periodically driven systems, such as shaken optical lattices with a
time-dependent shaking strength, a situation relevant to the cold atom
experiments.
