Existence of eigensets on bilinear control systems

Abstract

For bilinear control systems in ${\mathbb{R}}^d$ we prove, under an accessibility hypothesis, the existence of a nontrivial compact set $D\subset {\mathbb{R}}^d$ satisfying ${\mathcal{O}}_t(D)=e^{tR}D$ for all $t>0$, where $R\in \mathbb{R}$ is a fixed constant and ${\mathcal{O}}_t(D)$ denotes the orbit from $D$ at time $t$. This property generalizes the trajectory of an eigenvector on a linear dynamical system, and merits such a set the name "eigenset".