In this work, we define a verification procedure that enables to build guaranteed PGD-reduced models for linear elliptic or parabolic problems depending on many parameters. It is based on the general concept of constitutive relation error and provides for strict bounds on both global error and error on outputs of interest. Furthermore, it helps driving adaptive strategies by assessing contributions of various error sources. Consequently, virtual charts that may be constructed from the PGD approximate solution can be certified. Technicalities and performances of the control approach, in particular when dealing with a large set of model parameters, are detailed on a transient thermal problem.