On Differential-Algebraic Equations in Infinite Dimensions

TL;DR

This paper studies differential-algebraic equations in infinite-dimensional spaces, defining initial value spaces, solution types, and analyzing stability properties through spectral analysis of operator pencils.

Contribution

It introduces a framework for initial values and solutions of infinite-dimensional DAEs, and characterizes their stability using spectral properties.

Findings

Defined a space of consistent initial values for infinite-dimensional DAEs

Established existence of classical and mild solutions for arbitrary initial data

Characterized exponential stability and dichotomies via spectrum of operator pencils

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.