Optimal cross-over designs for total effects under a model with self and mixed carryover effects

We consider cross-over designs for a model that includes specific carryover effects when a treatment is preceded by itself. When the parameters of interest are total effects, i.e. the sum of direct effects of treatment and self-carryover effects, we show that optimal designs are a compromise between designs balanced on subjects such as balanced binary block designs and designs with subjects having a single treatment. We also propose universally optimal designs with a reduced number of subjects.