Existence of eigensets on bilinear control systems

TL;DR

This paper proves the existence of special invariant sets called eigensets in bilinear control systems, generalizing eigenvectors from linear systems, under certain accessibility conditions.

Contribution

It establishes the existence of eigensets for bilinear control systems, extending the concept of eigenvectors to nonlinear control dynamics.

Findings

Existence of eigensets under accessibility hypothesis

Eigensets satisfy a scaling property with exponential map

Generalizes eigenvector concept to bilinear 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".