Control sets of linear control systems on $\mathbb{R}^2$. The complex case

TL;DR

This paper explicitly characterizes the unique control set with non-empty interior for linear control systems on b2 with complex eigenvalues, showing its closure is bounded by a computable periodic orbit.

Contribution

It provides an explicit computation of the control set and its boundary for systems with complex eigenvalues, which was previously not explicitly described.

Findings

The control set's closure coincides with the region bounded by a periodic orbit.

The boundary of the control set is explicitly computable.

The control set is unique with non-empty interior for these systems.

Abstract

This paper explicitly computes the unique control set $D$ with non-empty interior of a linear control system on $\mathbb{R}^2$, when the associated matrix has complex eigenvalues. It turns out that the closure of $D$ coincides with the the region delimited by a computable periodic orbit $\mathcal{O}$ of the system.