A global Heymann´s Lemma for nonlinear systems

In linear system theory Heymann´s Lemma shows that a controllable multi-input system can be converted into a single-input controllable system by use of appropriate feedback. A global version of this lemma is developed. Two main results are obtained. The first result shows that if the (strong) controllability rank condition is satisfied generically by the original C_∞ system, then there exists C_∞ feedback (α, β) such that the closed loop single-input system satisfies the (strong) controllability rank condition generically. The second results gives a sufficient condition to ensure the existence of a C_∞ feedback such that the closed-loop single-input system satisfies the (strong) controllability rank condition globally. It is also shown that under some restrictions on the state manifold, such sufficient conditions are also necessary