Give an example of an experiment where a control group would not be necessary.

DoE studies, for example do not require control group, since they usually involve models in which the number of variables are substantially smaller than the number of experiments. The DoE analysis provides a description model that shows how the responses or output variables depend on the factor settings or input variables. As long as the unknowns are substantially less than the number of equations, there is very minimal risk of over-fitting.