On differential–algebraic equations in infinite dimensions

Sascha Trostorff, Marcus Waurick

Research output: Contribution to journalArticle

Abstract

We consider a class of differential–algebraic equations (DAEs) with index zero in an infinite dimensional Hilbert space. We define a space of consistent initial values, which lead to classical continuously differential solutions for the associated DAE. Moreover, we show that for arbitrary initial values we obtain mild solutions for the associated problem. We discuss the asymptotic behaviour of solutions for both problems. In particular, we provide a characterisation for exponential stability and exponential dichotomies in terms of the spectrum of the associated operator pencil.

Original languageEnglish
Number of pages36
JournalJournal of Differential Equations
Early online date23 Jul 2018
DOIs
Publication statusE-pub ahead of print - 23 Jul 2018

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Infinite Dimensions
Operator Pencil
Exponential Dichotomy
Mild Solution
Asymptotic Behavior of Solutions
Exponential Stability
Hilbert space
Hilbert spaces
Asymptotic stability
Zero
Arbitrary
Class

Keywords

  • consistent initial values
  • differential–algebraic equations
  • exponential dichotomy
  • exponential stability
  • strong and mild solutions

Cite this

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On differential–algebraic equations in infinite dimensions. / Trostorff, Sascha; Waurick, Marcus.

In: Journal of Differential Equations, 23.07.2018.

Research output: Contribution to journalArticle

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AU - Waurick, Marcus

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AB - We consider a class of differential–algebraic equations (DAEs) with index zero in an infinite dimensional Hilbert space. We define a space of consistent initial values, which lead to classical continuously differential solutions for the associated DAE. Moreover, we show that for arbitrary initial values we obtain mild solutions for the associated problem. We discuss the asymptotic behaviour of solutions for both problems. In particular, we provide a characterisation for exponential stability and exponential dichotomies in terms of the spectrum of the associated operator pencil.

KW - consistent initial values

KW - differential–algebraic equations

KW - exponential dichotomy

KW - exponential stability

KW - strong and mild solutions

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