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In this work, we propose to efficiently solve time dependent parametrizedoptimal control problems governed by parabolic partial differential equationsthrough the certified reduced basis method. In particular, we will exploit anerror estimator procedure, based on easy-to-compute quantities which guaranteea rigorous and efficient bound for the error of the involved variables. Firstof all, we propose the analysis of the problem at hand, proving itswell-posedness thanks to Ne\v{c}as - Babu\v{s}ka theory for distributed andboundary controls in a space-time formulation. Then, we derive error estimatorsto apply a Greedy method during the offline stage, in order to perform, duringthe online stage, a Galerkin projection onto a low-dimensional space spanned byproperly chosen high-fidelity solutions. We tested the error estimators on twomodel problems governed by a Graetz flow: a physical parametrized distributedoptimal control problem and a boundary optimal control problem with physicaland geometrical parameters. The results have been compared to a previouslyproposed bound, based on the exact computation of the Babu\v{s}ka inf-supconstant, in terms of reliability and computational costs. We remark that ourfindings still hold in the steady setting and we propose a brief insight alsofor this simpler formulation.

A Certified Reduced Basis Method for Linear Parametrized Parabolic Optimal Control Problems in Space-Time Formulation