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 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 G which is a semidirect product of a compact group K K and a vector space V V on which K K acts linearly. If F \mathcal {F} is a family of right-invariant vector fields, then the values of the elements of F \mathcal {F} at the identity define a subset Γ \Gamma of L ( G ) L(G) the Lie algebra of G G . We say that F \mathcal {F} is transitive on G G if the semigroup generated by ∪ X ∈ Γ { exp ⁡ ( t X ) : t ⩾ 0 } { \cup _{X \in \Gamma }}\{ \exp (tX):t \geqslant 0\} is equal to G G . Our main result is that F \mathcal {F} is transitive if and only if Lie ⁡ ( Γ ) \operatorname {Lie} (\Gamma ) , the Lie algebra generated by Γ \Gamma , is equal to L ( G ) L(G) .