论文标题
通过线性控制理论对微分 - 代数方程的几何分析
Geometric analysis of differential-algebraic equations via linear control theory
论文作者
论文摘要
我们考虑线性微分 - 代数方程式DAE和相应基质铅笔的Kronecker规范形式的KCF。我们还考虑线性控制系统及其Morse典型形式MCF。对于线性DAE，提出了一个名为equinlicitation的过程，该过程附加到任何线性dae上，定义为坐标更改，反馈转换和输出注入的线性控制系统。然后，我们将与DAE与DAE相关联的子空间与与控制系统相关的（也以几何方式）相关的（也以几何方式）的相关性进行比较，即，我们比较了DAE的Wong序列和控制系统的不变子空间。我们证明，线性DAE的KCF和控制系统的MCF具有完美的对应关系，并且它们的不变性是相关的。这样，我们将线性DAE的几何分析与经典的几何线性控制理论联系起来。最后，我们提出了一个名为DAE的内部等价概念，并讨论了其与内部规律性的关系，即解决方案的存在和唯一性。
We consider linear differential-algebraic equations DAEs and the Kronecker canonical form KCF of the corresponding matrix pencils. We also consider linear control systems and their Morse canonical form MCF. For a linear DAE, a procedure named explicitation is proposed, which attaches to any linear DAE a linear control system defined up to a coordinates change, a feedback transformation and an output injection. Then we compare subspaces associated to a DAE in a geometric way with those associated (also in a geometric way) to a control system, namely, we compare the Wong sequences of DAEs and invariant subspaces of control systems. We prove that the KCF of linear DAEs and the MCF of control systems have a perfect correspondence and that their invariants are related. In this way, we connect the geometric analysis of linear DAEs with the classical geometric linear control theory. Finally, we propose a concept named internal equivalence for DAEs and discuss its relation with internal regularity, i.e., the existence and uniqueness of solutions.
