On the Equivalence of Control Systems and the Linearization of Nonlinear Systems

Given two control systems where the control enters linearly, a necessary and sufficient condition is derived that these systems be locally diffeomorphic, i.e., that there exist a local diffeomorphism between the state spaces which carries a trajectory of the first system for each control into the trajectory of the second system for the same control. As a corollary we derive necessary and sufficient conditions for a system to be locally diffeomorphic to a linear system.