Feedback Classification of Invariant Control Systems on Three-Dimensional Lie Groups

We consider left-invariant control affine systems evolving on Lie groups. In this context, feedback equivalence specializes to detached feedback equivalence. We characterize (local) detached feedback equivalence in a simple algebraic manner. We then classify all (full-rank) systems evolving on three-dimensional Lie groups. A representative is identified for each equivalence class. Systems on the Heisenberg group, the Euclidean group, and the orthogonal group are treated in full, as typical examples. In these three cases, simple algebraic characterizations of the equivalence classes are also exhibited. A few remarks conclude the paper.