  External driving is emerging as a promising tool for exploring new phases in
quantum systems. The intrinsically non-equilibrium states that result, however,
are challenging to describe and control. We study the steady states of a
periodically driven one-dimensional electronic system, including the effects of
radiative recombination, electron-phonon interactions, and the coupling to an
external fermionic reservoir. Using a kinetic equation for the populations of
the Floquet eigenstates, we show that the steady-state distribution can be
controlled using the momentum and energy relaxation pathways provided by the
coupling to phonon and Fermi reservoirs. In order to utilize the latter, we
propose to couple the system and reservoir via an energy filter which
suppresses photon-assisted tunneling. Importantly, coupling to these reservoirs
yields a steady state resembling a band insulator in the Floquet basis. The
system exhibits incompressible behavior, while hosting a small density of
excitations. We discuss transport signatures, and describe the regimes where
insulating behavior is obtained. Our results give promise for realizing Floquet
topological insulators.
