Transitivity of Families of Invariant Vector Fields on The Semidirect Products of Lie Groups

In this paper we give necessary and sufficient conditions for a family of right (or left) invariant vector fields on a Lie group 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 which is a semidirect product of a compact group and a vector space on which acts linearly. If <!-- MATH $\mathcal{F}$ --> is a family of right-invariant vector fields, then the values of the elements of <!-- MATH $\mathcal{F}$ --> at the identity define a subset of the Lie algebra of . We say that <!-- MATH $\mathcal{F}$ --> is transitive on if the semigroup generated by <!-- MATH ${ \cup _{X \in \Gamma }}\{ \exp (tX):t \geqslant 0\}$ --> is equal to . Our main result is that <!-- MATH $\mathcal{F}$ --> is transitive if and only if <!-- MATH $\operatorname{Lie} (\Gamma )$ --> , the Lie algebra generated by , is equal to .