Spectral Characterizations of Solvability and Stability for Delay Differential-Algebraic Equations

TL;DR

This paper investigates the solvability and stability of delay differential-algebraic equations using spectral methods, introduces a new concept of weak stability, and highlights limitations of eigenvalue-based stability analysis.

Contribution

It provides spectral characterizations for solvability, demonstrates the restricted applicability of eigenvalue methods, and proposes weak stability for commuting DDAEs.

Findings

Spectral conditions characterize DDAE solvability.

Eigenvalue-based stability analysis is limited to non-advanced DDAEs.

Introduction of weak stability for commuting DDAEs.

Abstract

The solvability and stability analysis of linear time invariant systems of delay differential-algebraic equations (DDAEs) is analyzed. The behavior approach is applied to DDAEs in order to establish characterizations of their solvability in terms of spectral conditions. Furthermore, examples are delivered to demonstrate that the eigenvalue-based approach to analyze the exponential stability of dynamical system is only valid for a special class of DDAEs, namely non-advanced. Then, a new concept of weak stability is proposed and studied for DDAEs whose matrix coefficients pairwise commute.