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

TL;DR

This paper establishes conditions under which a linear control system on a solvable Lie group has a unique control set with a nonempty interior, linking algebraic properties to control set existence.

Contribution

It provides weak conditions involving the Lie algebra rank condition and compactness criteria that guarantee the existence and uniqueness of a control set with a nonempty interior.

Findings

Existence of control set under Lie algebra rank condition and compactness.

Uniqueness of the control set containing the generalized kernel.

Control set's closure includes the entire generalized kernel.

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.