Transitivity of linear control systems on Lie groups

Abstract

In this paper, we characterize the transitivity property of a linear control system Sigma on a connected Lie group G. The dynamic of Sigma is determined by the drift vector field which is an infinitesimal automorphism of G and by the control vectors which are elements of the Lie algebra L(G) of G. The Lie algebra rank condition which characterize transitivity does not determine controllability. We show this by an example on the simply connected nilpotent Heisenberg Lie group of dimension 3.