M\"obius transformations and the configuration space of a Hilbert snake

TL;DR

This paper simplifies the proof of controllability for a Hilbert snake by leveraging Möbius transformations on the configuration space within a separable Hilbert space, extending finite-dimensional results.

Contribution

It introduces a new approach using Möbius group actions to analyze controllability, generalizing finite-dimensional theorems to infinite-dimensional Hilbert spaces.

Findings

Simplified proof of Hilbert snake controllability

Extension of finite-dimensional theorems to infinite dimensions

Application of Möbius transformations to configuration spaces

Abstract

The purpose of this paper is to give a simpler proof to the problem of controllability of a Hilbert snake \cite{PeSa}. Using the action of the M\"obius group of the unit sphere on the configuration space, in the context of a separable Hilbert space. We give a generalization of the Theorem of accessibility contained in \cite{Ha} and \cite{Ro} for articulated arms and snakes in a finite dimensional Hilbert space