A characterization of the Lie algebra rank condition by transverse periodic functions

The Lie algebra rank condition plays a central role in nonlinear systems control theory. We show that the satisfaction of this condition by a set of smooth control vector fields is equivalent to the existence of smooth transverse periodic functions. The proof outlined-details can be found in Morin and Samson (2000)-is constructive and provides a method for the determination of such functions. This is illustrated by an example.