Controllability and semigroups of invariant control systems on ${\mathrm {Sl}}\left( n,{\mathbb {H}}\right)$

Abstract

Let ${\mathrm {Sl}}\left( n,{\mathbb {H}}\right)$ be the Lie group of $n\times n$ quaternionic matrices g with $\left| \det g\right| =1$. 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 special subgroup isomorphic to ${\mathrm {Sl}}\left( 2,{\mathbb {H}}\right)$. From this, we give sufficient conditions on $A,B\in \mathfrak {sl}\left( n, {\mathbb {H}}\right)$ to ensure 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 $\left( A,B\right) \in \mathfrak {sl}\left( n,{\mathbb {H}}\right) ^{2}$.