This paper considers the problem of controlled invariance of involutive regular distribution, both for smooth and real analytic cases. After a review of some existing work, a precise formulation of the problem of local and global controlled invariance of involutive regular distributions for both affine control systems and affine distributions is introduced. A complete characterization for local controlled invariance of involutive regular distributions for affine control systems is presented. A geometric interpretation for this characterization is provided. A result on local controlled invariance for real analytic affine distribution is given. Then, we investigate conditions that allow passages from local controlled invariance to global controlled invariance, for both smooth and real analytic affine distributions. We clarify existing results in the literature. Finally, for manifolds with a symmetry Lie group action, the problem of global controlled invariance is considered.

本文考虑了平稳和实际解析情况下对合正则分布的控制不变性问题。在回顾了一些现有工作之后，介绍了仿射控制系统和仿射分布的渐进正则分布的局部和全局控制不变性问题的精确表述。提出了仿射控制系统对合正则分布的局部控制不变性的完整表征。为此特征提供了几何解释。给出了真实解析仿射分布的局部控制不变性的结果。然后，我们研究了允许从局部受控不变性到全局受控不变性传递的条件，包括平滑和真实的仿射分布。我们澄清文献中的现有结果。最后，对于具有对称李群作用的流形，考虑了全局控制不变性的问题。