再论平凡化控制-仿射系统
Trivialisable control-affine systems revisited
摘要：
本文的目的是探讨平凡控制系统的概念，即系统的动态只取决于控制。Serres在控制-平面上具有标量控制的非线性系统的背景下引入并研究了平凡系统。在我们的工作中，我们首先提出了一个扩展的概念平凡的控制仿射系统的任意数量的国家和控制。接下来，我们的第一个结果涉及两个新的平凡控制仿射系统的特征，其中之一是基于无穷小对称性的研究，因此是几何的。其次，我们得到了无穷小对称的李代数具有传递的几乎交换李子代数的平凡控制仿射系统的标准形。第三，我们研究并给出了$3$维流形上平凡控制仿射系统的一个特征。特别是，我们给出了一个新的证明Serres获得的以前的特征。我们的特征是基于两个功能反馈不变量的属性：曲率（Agrachev介绍）和中心的仿射曲率（Wilkens使用）。最后，我们给出了控制仿射系统的几种规范形，它们的曲率和中心仿射曲率具有特殊的性质。
英文摘要：
The purpose of this paper is to explore the concept of trivial control systems, namely systems whose dynamics depends on the controls only. Trivial systems have been introduced and studied by Serres in the the context of control-nonlinear systems on the plane with a scalar control. In our work, we begin by proposing an extension of the notion of triviality to control-affine systems with arbitrary number of states and controls. Next, our first result concerns two novel characterisations of trivial control-affine systems, one of them is based on the study of infinitesimal symmetries and is thus geometric. Second, we derive a normal form of trivial control-affine systems whose Lie algebra of infinitesimal symmetries possesses a transitive almost abelian Lie subalgebra. Third, we study and propose a characterisation of trivial control-affine systems on $3$-dimensional manifolds with scalar control. In particular, we give a novel proof of the previous characterisation obtained by Serres. Our characterisation is based on the properties of two functional feedback invariants: the curvature (introduced by Agrachev) and the centro-affine curvature (used by Wilkens). Finally, we give several normal forms of control-affine systems, for which the curvature and the centro-affine curvature have special properties.
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