Existence and uniqueness of control sets with a nonempty interior for linear control systems on solvable groups

TL;DR

The paper addresses the existence and uniqueness of control sets with nonempty interior for linear control systems on solvable Lie groups. It develops a decomposition into the nilpotent component and the generalized kernel, and uses a Cartesian-product framework and automorphism-induced diffeomorphisms to analyze interior controllability. The main results show that if the Lie algebra rank condition holds and the nilpotent part of the generalized kernel is compact, there exists a unique control set with nonempty interior whose closure contains the generalized kernel, and this set corresponds to a preimage of a control set on a Cartesian model. This provides a robust structural characterization of interior controllability for linear control systems on solvable groups and a practical pathway to verify controllability via reductions to nilpotent and Cartesian components.

Abstract

In this paper, we obtain weak conditions for the existence of a control set with a nonempty interior for a linear control system on a solvable Lie group. We show that the Lie algebra rank condition together with the compactness of the nilpotent part of the generalized kernel of the drift are enough to assure the existence of such a control set. Moreover, this control set is unique and contains the whole generalized kernel in its closure.