基于非线性控制理论的非线性微分-代数方程几何分析
Geometric analysis of nonlinear differential-algebraic equations via nonlinear control theory
摘要：
对于非线性微分-代数方程，我们定义了两种等价，即外等价和内等价。粗略地说，“外部”一词意味着我们在任何地方都考虑DAE（局部），而“内部”意味着我们在其（局部）最大不变子流形上考虑DAE（即，其解决方案存在的地方）。首先，我们对微分代数方程求解理论中的几何约简方法进行了修正，并给出了一个可实现的算法。然后提出了一种连接非线性微分代数系统和非线性控制系统的方法--带驱动变量的显式化方法，并证明了显式化系统的驱动变量在一定的对合条件下可以约简。最后，由于显式，我们将使用一些概念，从非线性控制理论，以获得两个非线性推广的维尔斯特拉斯形式。
英文摘要：
For nonlinear differential-algebraic equations (DAEs), we define two kinds of equivalences, namely, the external and internal equivalence. Roughly speaking, the word "external" means that we consider a DAE (locally) everywhere and "internal" means that we consider the DAE on its (locally) maximal invariant submanifold (i.e., where its solutions exist) only. First, we revise the geometric reduction method in DAEs solution theory and formulate an implementable algorithm to realize that method. Then a procedure named explicitation with driving variables is proposed to connect nonlinear DAEs with nonlinear control systems and we show that the driving variables of an explicitation system can be reduced under some involutivity conditions. Finally, due to the explicitation, we will use some notions from nonlinear control theory to derive two nonlinear generalizations of the Weierstrass form.
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