Identifying a Distinguished Control in Dynamic Feedback Linearization and in Reduction

In this paper we discuss two important problems in control theory: a) dynamic feedback linearization via successive prolongations of a suitably chosen control, and b) linearization via a one-fold reduction. Despite their differences, both problems require identifying a special control, called distinguished control, which is a smooth combination of the original controls of the system. In each case, this control is determined with the help of a corank-one suitable subdistribution of D0, the distribution spanned by the original control vector fields. The main result of the paper provides a direct explicit construction of that corank-one distinguished distribution and of the distinguished control.