Controllability of systems on solvable Lie groups: the generic case
A solvable Lie algebra \({\mathfrak g}\) is called \(I_nD_m\)-controllable if there exist elements \(A_1,\dots, A_m\), \(B_1,\dots,\) \(B_n\in{\mathfrak g}\) such that the set \((\bigcup j= 1^m \exp\mathbb{R}^+ A_j)\cup(\bigcup k= 1^n \exp\mathbb{R} B_k)\) generates the simply connected Lie group \(G\) with Lie algebra \({\mathfrak g}\). The author shows that there is a minimal \(r\) such that \({\mathfrak g}\) is \(I_rD_0\)-controllable (and then generic \(B_k\) will do). This \(r\) is called the generic controllability rank. He further shows that there also is a minimal \(i\leq r\) such that \({\mathfrak g}\) is \(I_iD_d\)-controllable with \(i+d= r\) (again this is then generic in the \(A_j\) and the \(B_k\)). Finally, he gives a concrete way to determine \(r\) and \(i\) from the structure of \({\mathfrak g}\). It is based on an a priori reduction to metabelian Lie algebras.
scientific article; zbMATH DE number 1901423