Index Concepts for Differential-Algebraic Equations

Introduction

Differential-algebraic equations (DAEs) present today the state of the art in mathematical modeling of dynamical systems in almost all areas of science and engineering. Modeling is done in a modularized way by combining standardized sub-models in a hierarchically built network. The topic is well studied from an analytical, numerical, and control theoretical point of view, and several monographs are available that cover different aspects of the subject [1, 2, 9, 14–16, 21, 28, 29, 34].

The mathematical model can usually be written in the form

$$\displaystyle{ F(t,x,\dot{x}) = 0, }$$

(1)

where $\dot{x}$ denotes the (typically time) derivative of x. Denoting by $C^{k}(\mathbb{I}, \mathbb{R}^{n})$ the set of k times continuously differentiable functions from $\mathbb{I} = [\underline{t},\overline{t}] \subset \mathbb{R}$ to $\mathbb{R}^{n}$, one usually assumes that $F \in C^{0}(\mathbb{I} \times \mathbb{D}_{x} \times \mathbb{D}_{\dot{x}}, \mathbb{R}^{m})$is...