Conjugation of unique and non-unique parts of optimal control

Optimization problems linear for scalar control are discussed. If the optimal path originating from a certain point contains a stretch of optimal control all optimal paths from proximate points are shown to feature this property. The non-unique and the unique stretches are conjugated (for all paths with an even order of the unique control $q$ and for almost all of them with odd $q>1$) by an infinite number of control switchings.