On Sussmann theorem for orbits of sets of vector fields on Banach manifolds

The purpose of this paper is to give some generalizations, in the context of Banach manifolds, of Sussmannʼs results about the orbits of families of vector fields (Sussmann, 1973 [16]). Essentially, we define the notion of “l1-orbits” for any family of vector fields on a Banach manifold, and we prove, under appropriate assumptions, that such an orbit is a weak Banach submanifold.