Control systems on nilpotent Lie groups of dimension ≤4: Equivalence and classification

Left-invariant control affine systems on simply connected nilpotent Lie groups of dimension ≤4 are considered. First, we classify these control systems under two natural equivalence relations. Second, the associated cost-extended systems (with invariant quadratic Lagrangian) are classified. The latter classification is reinterpreted in terms of invariant affine distributions equipped with quadratic cost; also, we obtain a classification of the invariant sub-Riemannian (and Riemannian) structures. A few remarks conclude the paper.