Continuous dependence of the controllability set of a nonlinear control system on a parameter

In this note, we consider the mapping which associates a controllability set to each value of the parameter of a system. Sufficient conditions for the existence of breaks of this mapping are obtained. They show that nearly all such mappings have a finite number of breaks.This note was written during the author's stay at the University of Hamburg. He expresses his gratitude to the Institut für Angewandte Mathematik for its warm hospitality.