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

Tomasz Goliński, Fernand Pelletier

TL;DR

This work develops a comprehensive Banach-manifold framework for regulated curves on Banach manifolds, defining $k$-regulated paths in both spaces and bundles. By constructing atlas-based coordinates and leveraging pull-back representations, it proves that spaces of $1$-regulated paths and their $0$-regulated lifts carry natural Banach structures, with the projection to the base path space forming a Banach bundle. The results generalize prior finite-dimensional and strongly Riemannian cases, enabling global analysis of endpoint maps and controllability in infinite-dimensional settings. The framework lays the groundwork for studying singularities of the endpoint map in Regulated curve settings, with potential applications to infinite-dimensional sub-Riemannian-type problems and control theory.

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.

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

TL;DR

This work develops a comprehensive Banach-manifold framework for regulated curves on Banach manifolds, defining -regulated paths in both spaces and bundles. By constructing atlas-based coordinates and leveraging pull-back representations, it proves that spaces of -regulated paths and their -regulated lifts carry natural Banach structures, with the projection to the base path space forming a Banach bundle. The results generalize prior finite-dimensional and strongly Riemannian cases, enabling global analysis of endpoint maps and controllability in infinite-dimensional settings. The framework lays the groundwork for studying singularities of the endpoint map in Regulated curve settings, with potential applications to infinite-dimensional sub-Riemannian-type problems and control theory.

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.
Paper Structure (9 sections, 13 theorems, 42 equations)

This paper contains 9 sections, 13 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. $k$-regulated paths in a Banach space
  3. $k$-regulated paths in a Banach manifold
  4. $0$-regulated lifts of $1$-regulated paths
  5. Banach structure on the set of $0$-regulated sections and $0$-regulated vector fields along $1$-regulated paths
  6. Topology on the set of regulated lifts of paths
  7. Banach manifold structure of $\mathcal{P}(E)$
  8. Banach manifold structure in the case of $C^1$ curves
  9. Banach structure on $\mathcal{C}(E)$

Key Result

Theorem 1

${}$

Theorems & Definitions (26)

  • Theorem 1
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Theorem 2.6
  • Proposition 2.7
  • Proposition 2.8
  • Proposition 2.9
  • Definition 3.1
  • ...and 16 more