A General Theorem on Local Controllability

We prove a general sufficient condition for local controllability of a nonlinear system at an equilibrium point. Earlier results of Brunovsky, Hermes, Jurdjevic, Crouch and Byrnes, Sussmann and Grossmann, are shown to be particular cases of this result. Also, a number of new sufficient conditions are obtained. All these results follow from one simple general principle, namely, that local controllability follows whenever brackets with certain symmetries can be “neutralized,” in a suitable way, by writing them as linear combinations of brackets of a lower degree. Both the class of symmetries and the definition of “degree” can be chosen to suit the problem.