Approximately controllable finite-dimensional bilinear systems are controllable

TL;DR

This paper establishes a geometric characterization of approximate controllability in finite-dimensional bilinear systems, linking it to controllability in the punctured space and extending results to homogeneous systems.

Contribution

It provides a geometric criterion for approximate controllability of bilinear systems and extends the analysis to angularly controllable homogeneous systems.

Findings

Approximate controllability is equivalent to controllability in rac{ }{n}rom the origin.

No codimension-one foliation with dense leaves exists in rac{ }{n}or the system.

The geometric approach extends controllability results to homogeneous systems.

Abstract

We show that a bilinear control system is approximately controllable if and only if it is controllable in $\mathbb{R}^{n}\setminus\{0\}$. We approach this problem by looking at the foliation made by the orbits of the system, and by showing that there does not exist a codimension-one foliation in $\mathbb{R}^{n}\setminus\{0\}$ with dense leaves that are everywhere transversal to the radial direction. The proposed geometric approach allows to extend the results to homogeneous systems that are angularly controllable.