Controllability of linear systems on solvable Lie groups

TL;DR

This paper investigates the controllability of linear systems on solvable Lie groups, establishing conditions based on eigenvalues of associated derivations and the openness of the reachable set.

Contribution

It provides new controllability criteria for linear systems on solvable Lie groups, linking eigenvalues and the structure of the reachable set.

Findings

Controllability is guaranteed if the reachable set of the neutral element is open and derivation D has only pure imaginary eigenvalues.

For bounded systems on nilpotent Lie groups, these conditions are also necessary.

The study extends classical controllability results from Euclidean spaces to more general Lie group settings.

Abstract

Linear systems on Lie groups are a natural generalization of linear system on Euclidian spaces. For such systems, this paper studies controllability by taking in consideration the eigenvalues of an associated derivation D. When the state space is a solvable connected Lie group, controllability of the systems is guaranteed if the reachable set of the neutral element is open and the derivation D has only pure imaginary eigenvalues. For bounded systems on nilpotent Lie groups such conditions are also necessary.