Transitivity of families of invariant vector fields on the semidirect products of Lie groups

Abstract:

In this paper we give necessary and sufficient conditions for a family of right (or left) invariant vector fields on a Lie group $G$ to be transitive. The concept of transitivity is essentially that of controllability in the literature on control systems. We consider families of right (resp. left) invariant vector fields on a Lie group $G$ which is a semidirect product of a compact group $K$ and a vector space $V$ on which $K$ acts linearly. If $\mathcal {F}$ is a family of right-invariant vector fields, then the values of the elements of $\mathcal {F}$ at the identity define a subset $\Gamma$ of $L(G)$ the Lie algebra of $G$. We say that $\mathcal {F}$ is transitive on $G$ if the semigroup generated by ${ \cup _{X \in \Gamma }}\{ \exp (tX):t \geqslant 0\}$ is equal to $G$. Our main result is that $\mathcal {F}$ is transitive if and only if $\operatorname {Lie} (\Gamma )$, the Lie algebra generated by $\Gamma$, is equal to $L(G)$.