  We investigate a long time asymptotic state of periodically driven open
quantum systems analytically. The model we consider in this paper is a free
fermionic system coupled to an energy and particle reservoir. We clarify some
generic properties of the system which are independent of the details of the
reservoir in the high frequency regime of the external driving. When the
frequency of the external driving is much larger than the energy cutoff of the
system-reservoir coupling, the low-energy properties of the system are
equivalent to those of the Gibbs distribution of a Floquet effective
Hamiltonian. Furthermore, we investigate the effect of finite dissipation on
the system, and elucidate that we cannot suppress excitations by merely
increasing the system-reservoir coupling because the excitations are created
through the reservoir by the external driving when the frequency is smaller
than the energy cutoff of the system-reservoir coupling.
