Controllability of linear systems on Heisenberg groups ℍⁿ

Abstract

This paper is devoted to the study of global controllability of linear systems on higher dimensional Heisenberg groups ${\mathbb{H}}^n$. Necessary conditions and sufficient ones are provided. They are compared to the known necessary and sufficient conditions on ${\mathbb{H}}^1$ stated in Dath and Jouan [(2016). Controllability of linear systems on low dimensional nilpotent and solvable Lie groups. Journal of Dynamical and Control Systems, 22(2), 207–225]. The notion of decoupled systems is introduced in order to get examples and counter-examples, and more precise controllability criteria are stated for them.

Keywords:

• Nilpotent Lie groups
• Heisenberg groups
• linear systems
• controllability

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 In other words the identity e should belong to an open subset of G included in the set of reachable points from e in time less than or equal to t with inputs uniformly bounded by B.

2 The derived subalgebras ${\mathcal{D}}^k\mathfrak{g}$ are defined inductively by ${\mathcal{D}}^0\mathfrak{g}=\mathfrak{g}$ and ${\mathcal{D}}^{k+1}\mathfrak{g}=[\mathfrak{g},{\mathcal{D}}^k\mathfrak{g}]$.

3 Because ${\mathbb{H}}^n$ is a simply connected nilpotent Lie group.