Analysis of linear variable coefficient delay differential-algebraic equations
type: Item
instance of: scholarly article
title: Analysis of linear variable coefficient delay differential-algebraic equations
zbMATH Open document ID: 1325.34084
author: Phi Ha
author: Andreas Steinbrecher
author: Volker Mehrmann
published in: Journal of Dynamics and Differential Equations
publication date: 2015-01-23T00:00:00Z
full work available at URL: https://depositonce.tu-berlin.de/handle/11303/15779
review text: This article studies delay-differential algebraic equations (DDAEs) described by \[ A \dot{x}(t) + B x(t) + C x(t-\tau) = f, \] where \(A\), \(B\), \(C\) and \(f\) are time-varying matrices and a vector. Explicit DDAEs, given by \[ \begin{align*}{ \dot{x}_1(t) &= D x (t) + E x(t-\tau) + f_1 \cr x_2(t) &= F x_1(t) + G x(t-\tau) + f_2 \cr}\end{align*}, \; x = (x_1, x_2), \] are a well-known subset of this class which encompasses -- the explicit versions of -- delay-differential equations, neutral equations and difference equations. For explicit DDAEs, the initial value problem is well-posed given appropriate consistency conditions for many functional spaces of solutions (see e.g. [\textit{M. C. Delfour} and \textit{J. Karrakchou}, J. Math. Anal. Appl. 125, 361--399, 400--450 (1987; Zbl 0643.93034)] for the time-invariant case). On the contrary, the original system -- or implicit -- may be over or underdetermined, hence the analysis of the initial value problem is more complex. Sometimes these patterns are obvious (the matrices \(A\), \(B\) and \(C\) are not necessarily square or nonsingular), but they may also result from an implicit \textit{redundancy} between the equations. This article describes a step-by-step methodology to explicit the system redundancies, using only linear transformations, differentiations and time shifts. This general process is first exposed for differential-algebraic equations and difference equations; in the simplest cases, the transformation leads to a system with the structure \[ \left[ \begin{matrix} A \\ 0 \\ 0 \end{matrix}\right] \dot{x}(t) + \left[ \begin{matrix} ? \\ {B} \\ 0 \end{matrix}\right] x(t) + \left[ \begin{matrix} ? \\ ? \\ 0 \end{matrix} \right] x(t-\tau) = \left[ \begin{matrix} f_1 \\ f_2 \\ f_3 \end{matrix} \right], \] where the absence of implicit redundancy is characterized by a matrix \([A^t, B^t]^t\) of full row rank. Note that when additionally this matrix is square and \(f_3\) is an empty vector, the system -- that …
reviewed by: Sébastien Boisgérault
Mathematics Subject Classification ID: 34K32
Mathematics Subject Classification ID: 34K17
Mathematics Subject Classification ID: 34K06
zbMATH DE Number: 6389979
zbMATH Keywords: differential algebraic equations
zbMATH Keywords: delay-differential-algebraic equations
zbMATH Keywords: existence of solutions
zbMATH Keywords: uniqueness of solutions
zbMATH Keywords: consistency conditions
describes a project that uses: RODAS
MaRDI profile type: Publication
OpenAlex ID: W2010676435
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DOI: 10.1007/S10884-014-9386-X
description: scientific article; zbMATH DE number 6389979