Control Systems on the Engel Group

Abstract

We consider control affine systems, as well as cost-extended control systems, on the (four-dimensional) Engel group. Specifically, we classify the full-rank left-invariant control affine systems (under both detached feedback equivalence and strongly detached feedback equivalence). The cost-extended control systems with quadratic cost are then classified (under cost equivalence), as are their associated Hamilton-Poisson systems (up to affine isomorphism). In all cases, we exhibit a complete list of equivalence class representatives.