We consider an experiment with blocks of equal size involving some qualitative treatments, one of which is a control treatment, and some controlled explanatory variables that can vary only from one block to another. The assumed model includes interactions between treatments and explanatory variables. The objective of the experiment is to compare the mean responses of the treatments with the mean response of the control for all possible values of the explanatory variables. First, conditions for a design to be optimal are established in the set-up of exact design theory. Then, the construction of designs that satisfy these conditions is studied. Conditions for a design to be optimal under several models are also given. Finally, an example is presented.