Control sets of one-input linear control systems on solvable, nonnilpotent 3D Lie groups

TL;DR

This paper fully characterizes the control sets of one-input linear control systems on certain 3D Lie groups, revealing conditions for their existence, uniqueness, and structure, and providing a comprehensive classification.

Contribution

It offers a complete description of control sets on solvable, nonnilpotent 3D Lie groups, including conditions for their existence and structure, advancing the understanding of control systems on these groups.

Findings

Control sets are unique and cylindrical when the derivation's restriction is nontrivial.

Multiple control sets with empty interiors can occur when the restriction is trivial.

Controllability depends on the group and the derivation's properties.

Abstract

In this article, we completely describe the control sets of one-input linear control systems on solvable, nonnilpotent 3D Lie groups. We show that, if the restriction of the associate derivation to the nilradical is nontrivial, the Lie algebra rank condition is enough to assure the existence of a control set with a nonempty interior. Moreover, such a control set is unique and, up to conjugations, given as a cylinder of the state space. On the other hand, if such a restriction is trivial, one can obtain an infinite number of control sets with empty interiors or even controllability, depending on the group considered.