Note on geometric algebras and control problems with SO(3)-symmetries

TL;DR

This paper explores the use of geometric algebra to analyze control problems with SO(3) symmetries, providing new insights into geodesics and local control algorithms on Carnot groups of step 2.

Contribution

It introduces a geometric algebra framework for control problems with SO(3) symmetry, offering a novel approach to understanding geodesics and control algorithms.

Findings

Geodesics are characterized as curves in geometric algebras.

A new algorithm for local control is proposed.

Application to Carnot groups of step 2 with SO(3) symmetry.

Abstract

We study the role of symmetries in control systems through the geometric algebra approach. We discuss two specific control problems on Carnot groups of step $2$ invariant with respect to the action of $SO(3)$. We understand the geodesics as the curves in suitable geometric algebras which allows us to assess a new algorithm for the local control.