A general theorem of symmetries and local controllability

We state 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, Grossman, turn out to be particular cases of this result. Also, a number of new sufficient conditions can be derived. All these results are consequences of 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.