  In this work, we propose to efficiently solve time dependent parametrized
optimal control problems governed by parabolic partial differential equations
through the certified reduced basis method. In particular, we will exploit an
error estimator procedure, based on easy-to-compute quantities which guarantee
a rigorous and efficient bound for the error of the involved variables. First
of all, we propose the analysis of the problem at hand, proving its
well-posedness thanks to Ne\v{c}as - Babu\v{s}ka theory for distributed and
boundary controls in a space-time formulation. Then, we derive error estimators
to apply a Greedy method during the offline stage, in order to perform, during
the online stage, a Galerkin projection onto a low-dimensional space spanned by
properly chosen high-fidelity solutions. We tested the error estimators on two
model problems governed by a Graetz flow: a physical parametrized distributed
optimal control problem and a boundary optimal control problem with physical
and geometrical parameters. The results have been compared to a previously
proposed bound, based on the exact computation of the Babu\v{s}ka inf-sup
constant, in terms of reliability and computational costs. We remark that our
findings still hold in the steady setting and we propose a brief insight also
for this simpler formulation.
