Regulated curves on a Banach manifold and singularities of endpoint map. I. Banach manifold structure

TL;DR

This paper constructs a Banach manifold structure for regulated curves in Banach bundles, extending previous results to general Banach manifolds, and discusses controllability issues related to these curves.

Contribution

It generalizes the Banach manifold structure of regulated curves from Riemannian to arbitrary Banach manifolds, using a new proof approach.

Findings

Established a Banach manifold structure for regulated curves in any Banach manifold.

Extended previous results from Riemannian to general Banach manifolds.

Discussed controllability problems related to these curves.

Abstract

We consider regulated curves in a Banach bundle whose projection on the basis is continuous with regulated derivative. We build a Banach manifold structure on the set of such curves. This result was previously obtained for the case of strong Riemannian Banach manifold and absolutely continuous curves in arXiv:1612.02604. The essential argument used was the existence of a "local addition" on such a manifold. Our proof is true for any Banach manifold. In the second part of the paper the problems of controllability will be discussed.