Controllability of hypersurface and solvable invariant systems (Q1972709)

Let \(G\) be a connected Lie group, \(L\) its Lie algebra, and \(A, B_1, \dots, B_m\in L\) considered as right invariant vector fields on \(G\). Consider the set \(\Gamma=A+ \sum^m_{j=1} \mathbb{R} B_j\) and the subsemigroup \(S\) generated by the \(\exp\mathbb{R}_+X\) with \(X\in\Gamma\). The set \(\Gamma\) is referred to as an (invariant) affine control system, which is called controllable if \(S=G\). Let \(L_0\) be the Lie subalgebra generated by \(B_1,\dots,B_m\) and \(G_0\) the corresponding analytic subgroup of \(G\). Suppose that \(\text{codim} L_0=1\). Then the author shows that \(\Gamma\) is controllable iff \(A\notin L_0\) (plus the condition \(G/G_0 \cong S^1\) in case \(G_0\) is closed). If \(G\) is simply connected and solvable such that all elements of \(\operatorname {ad}L\) have real spectrum, then \(\Gamma\) is controllable iff \(L=L_0\).