Local controllability about a reference trajectory

In this paper we extend a technique for providing sufficient conditions for local controllability about an equilibrium, to the problem of local controllability about a reference trajectory. The technique employs the existence of symmetries, as in a previous paper by P. E. Crouch and C. I. Byrnes, and more generally by H. J. Sussman, to provide specific variations along the reference trajectory not usually considered in either this problem, or the problem of finding necessary conditions in Optimal Control Theory.