Approximately controllable finite-dimensional bilinear systems are controllable

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.

9 pages, 2 figures