# Control sets of linear systems on semi-simple Lie groups

## TL;DR

This paper investigates the properties of control sets with nonempty interior for linear control systems on semisimple Lie groups, revealing differences from solvable cases and describing their structure.

## Contribution

It characterizes the structure and number of control sets with nonempty interior for linear systems on semisimple Lie groups, highlighting key differences from solvable groups.

## Key findings

- Multiple control sets with nonempty interior can exist on semisimple Lie groups.
- Control sets are contained in right translations of a fundamental control set.
- Distinct from solvable cases, the number of control sets can be greater than one.

## Abstract

In this paper we study the main properties of control sets with nonempty interior of linear control systems on semisimple Lie groups. We show that, unlike the solvable case, linear control systems on semisimple Lie groups may have more than one control set with nonempty interior and that they are contained in right translations of the one around the identity.