Semigroups and Controllability of Invariant Control Systems on $\mathrm{Sl}\left(n,\mathbb{H}\right)$

TL;DR

This paper investigates the structure of subsemigroups in the quaternionic special linear group and provides conditions under which certain control systems on this group are controllable, showing these conditions are generically satisfied.

Contribution

It establishes a semigroup generation result for $ ext{Sl}(n, ext{H})$ and derives generic controllability conditions for invariant control systems on this group.

Findings

Subsemigroup with nonempty interior containing $ ext{Sl}(2, ext{H})$ generates $ ext{Sl}(n, ext{H})$.

Sufficient controllability conditions are provided for systems on $ ext{Sl}(n, ext{H})$.

Controllability conditions are generic, forming an open and dense set in the parameter space.

Abstract

Let $\mathrm{Sl}\left( n,\mathbb{H}\right)$ be the Lie group of $n\times n$ quaternionic matrices $g$ with $\left\vert \det g\right\vert =1$. We prove that a subsemigroup $S \subset \mathrm{Sl}\left( n,\mathbb{H}\right)$ with nonempty interior is equal to $\mathrm{Sl}\left( n,\mathbb{H}\right)$ if $S$ contains a subgroup isomorphic to $\mathrm{Sl}\left( 2,\mathbb{H}\right)$. As application we give sufficient conditions on $A,B\in \mathfrak{sl}\left( n,\mathbb{H}\right)$ to ensuring that the invariant control system $\dot{g}=Ag+uBg$ is controllable on $\mathrm{Sl}\left( n,\mathbb{H}\right)$. We prove also that these conditions are generic in the sense that we obtain an open and dense set of controllable pairs $\left( A,B\right)\in\mathfrak{sl}\left( n,\mathbb{H}\right)^{2}$.