Adjoint Pairs of Differential-Algebraic Equations and Their Lyapunov Exponents
Adjoint Pairs of Differential-Algebraic Equations and Their Lyapunov Exponents
复制标题
微分代数方程的伴随对及其李亚普诺夫指数
DOI：
--
复制
发表时间：
2017
期刊：
影响因子：
--
通讯作者：
R. März
中科院分区：
文献类型：
--
作者：
V. H. Linh;R. März
本文研究了任意高易处理指数的正则微分代数方程的伴随对。我们考虑标准形式的DAE和DAE适当涉及的衍生物。我们介绍了因式分解伴随对的概念，并展示了它们的共同结构，包括索引和特征值。我们精确地描述了所谓的固有显式正则常微分方程（IERODE）和正则DAE的基本基本常微分方程（EUODEs）之间的关系。我们证明了在一对伴随的正则DAE的EUODE之间总是有那些彼此伴随的。此外，我们推广的李雅普诺夫指数理论的DAE具有任意高的指标，并建立了一般类的DAE是正规的李雅普诺夫意义。Perron恒等式在常微分方程理论中是众所周知的，但对于李雅普诺夫正则微分方程的伴随对一般不成立。我们建立标准的Perron身份是有效的。文中还举例说明了新的结果。
This paper is devoted to the analysis of adjoint pairs of regular differential-algebraic equations with arbitrarily high tractability index. We consider both standard form DAEs and DAEs with properly involved derivative. We introduce the notion of factorization-adjoint pairs and show their common structure including index and characteristic values. We precisely describe the relations between the so-called inherent explicit regular ODE (IERODE) and the essential underlying ODEs (EUODEs) of a regular DAE. We prove that among the EUODEs of an adjoint pair of regular DAEs there are always those which are adjoint to each other. Moreover, we extend the Lyapunov exponent theory to DAEs with arbitrarily high index and establish the general class of DAEs being regular in Lyapunov’s sense. The Perron identity which is well known in the ODE theory does not hold in general for adjoint pairs of Lyapunov regular DAEs. We establish criteria for the Perron identity to be valid. Examples are also given for illustrating the new results.