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

TL;DR

This paper generalizes Sussmann's theorem to Banach manifolds by defining l1-orbits for families of vector fields and proving these orbits form weak Banach submanifolds under certain conditions.

Contribution

It introduces the concept of l1-orbits on Banach manifolds and establishes their structure as weak Banach submanifolds, extending prior finite-dimensional results.

Findings

l1-orbits are weak Banach submanifolds

generalization of Sussmann's theorem to Banach manifolds

under appropriate assumptions, orbits have a manifold structure

Abstract

The purpose of this paper is to give some generalizations, in the context of Banach mani- folds, of Sussmann's results about the orbits of families of vector fields ([Su]). 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.