Sub-Riemannian Geometry and Geodesics in Banach Manifolds

TL;DR

This paper extends sub-Riemannian geometry to Banach manifolds, establishing controllability results and conditions for Hamiltonian geodesic flows in infinite dimensions, despite the absence of Pontryagin's principle.

Contribution

It introduces sub-Riemannian structures on Banach manifolds and generalizes key theorems for controllability and geodesic existence in infinite-dimensional spaces.

Findings

Extended Chow-Rashevski theorem for Banach manifolds

Provided conditions for Hamiltonian geodesic flow existence

Addressed challenges due to lack of Pontryagin Maximum Principle

Abstract

In this paper, we define and study sub-Riemannian structures on Banach manifolds. We obtain extensions of the Chow-Rashevski theorem for exact controllability, and give conditions for the existence of a Hamiltonian geodesic flow despite the lack of a Pontryagin Maximum Principle in the infinite dimensional setting.