We show that a bilinear control system is approximately controllable if and only if it is controllable in ${\mathbb{R}}^n\setminus \left\{0\right\}$. We approach this property 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 ${\mathbb{R}}^n\setminus \left\{0\right\}$ 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.

中文翻译：

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近似可控的有限维双线性系统是可控的

我们证明了双线性控制系统是近似可控的当且仅当它在 ${\mathbb{电阻}}^n\setminus \left\{0\right\}$. 我们通过观察由系统的轨道形成的叶理，并证明不存在共维一个叶理来接近这个性质。${\mathbb{电阻}}^n\setminus \left\{0\right\}$密集的叶子到处都是横向于径向方向。所提出的几何方法允许将结果扩展到角度可控的均匀系统。