  We study the dynamics of correlation functions of a class of $d-$dimensional
integrable models coupled linearly to a fermionic or bosonic bath in the
presence of a periodic drive with a square pulse protocol. It is well known
that in the absence of the bath, these models exhibit a dynamical phase
transition; all correlators decay to their steady state values as
$n_0^{-(d+1)/2}$[$n_0^{-d/2}]$ above [below] a critical frequency $\omega_c$,
where $n_0$ is the number of drive cycles. We find that the presence of a
linearly coupled fermionic bath which maintains integrability of the system
preserves this transition. We provide a semi-analytic expression for the
evolution operator for this system and use it to provide a phase diagram
showing the different dynamical regimes as a function of the system-bath
coupling strength and the bath parameters. In contrast, when such models are
coupled to a bosonic bath which breaks integrability of the model, we find
exponential decay of the correlators to their steady state. Our numerical
analysis shows that this exponential decay sets in above a critical number of
drive cycles $n_c$ which depends on the system-bath coupling strength and the
amplitude of perturbation. Below $n_c$, the system retains the power-law
behavior identical to that for the closed integrable models and the dynamical
transition survives. We discuss the applicability of our results for
interacting fermion systems and discuss experiments which can test our theory.
