  We compute the Floquet Hamiltonian $H_F$ for weakly interacting fermions
subjected to a continuous periodic drive using a Floquet perturbation theory
(FPT) with the interaction amplitude being the perturbation parameter. This
allows us to address the dynamics of the system at intermediate drive
frequencies $\hbar \omega_D \ge V_0 \ll {\mathcal J}_0$, where ${\mathcal J}_0$
is the amplitude of the kinetic term, $\omega_D$ is the drive frequency, and
$V_0$ is the typical interaction strength between the fermions. We compute, for
random initial states, the fidelity $F$ between wavefunctions after a drive
cycle obtained using $H_F$ and that obtained using exact diagonalization (ED).
We find that FPT yields a substantially larger value of $F$ compared to its
Magnus counterpart for $V_0\le \hbar \omega_D$ and $V_0\ll {\mathcal J}_0$. We
use the $H_F$ obtained to study the nature of the steady state of an weakly
interacting fermion chain; we find a wide range of $\omega_D$ which leads to
subthermal or superthermal steady states for finite chains. The driven
fermionic chain displays perfect dynamical localization for $V_0=0$; we address
the fate of this dynamical localization in the steady state of a finite
interacting chain and show that there is a crossover between localized and
delocalized steady states. We discuss the implication of our results for
thermodynamically large chains and chart out experiments which can test our
theory.
