Existence of eigensets for bilinear control systems

For bilinear control systems in $ \mathbb{R}^d $, we prove under certain assumptions 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 in a linear dynamical system, and such a set merits the name 'eigenset'.