A Semigroup associated to a linear control system on a Lie group

TL;DR

This paper introduces a new semigroup structure associated with linear control systems on Lie groups, enabling algebraic analysis and new controllability insights.

Contribution

It defines a novel semigroup S linked to the control system, facilitating the application of semigroup theory to control problems on Lie groups.

Findings

The semigroup S provides a new framework for analyzing controllability.

The approach yields new controllability results for linear control systems.

The algebraic structure simplifies the study of accessibility sets.

Abstract

Let us consider a linear control system \Sigma on a connected Lie group G. It is known that the accessibility set A from the identity e is in general not a semigroup. In this article we associate a new algebraic object S to \Sigma which turns out to be a semigroup, allowing the use of the semigroup machinery to approach \Sigma. In particular, we obtain some controllability results.