A complete classification of control sets for singular linear control systems on the Heisenberg group

TL;DR

The paper solves the singular LCS controllability problem on the Heisenberg group by reducing systems to normal forms via automorphisms and examining the invariants $\det A$ and $\operatorname{tr} A$ of the planar submatrix. It provides a complete classification of control sets under the $\operatorname{LARC}$, distinguishing three regimes: $(i)$ $\det A=\operatorname{tr} A=0$, where either global controllability or a continuum of equilibria occurs; $(ii)$ $\det A\neq 0$, $\operatorname{tr} A=0$, where admissibility is governed by the ad-rank condition $\alpha$ and eigenstructure, yielding cylinders or lines of one-point control sets; and $(iii)$ $\det A=0$, $\operatorname{tr} A\neq 0$, where the unique control set is the preimage of a planar affine-control set. This work not only completes the singular case on $\mathbb{H}$ but also provides a robust framework for extending control-set classifications to higher-dimensional nilpotent Lie groups. The results have implications for global controllability and design of control strategies on non-Euclidean spaces.

Abstract

In this paper, we investigate the control sets of linear control systems on the Heisenberg group associated with singular derivations. Under the Lie algebra rank condition, we provide a complete characterization of these sets by analyzing the trace and determinant of an associated 2 \times 2 submatrix.