A pseudo-resolvent approach to abstract differential-algebraic equations

TL;DR

This paper introduces a novel pseudo-resolvent approach for analyzing the solvability of linear abstract differential-algebraic equations using degenerate semigroups, with applications to electrical circuits and heat-wave systems.

Contribution

It presents a new index concept based on polynomial growth of pseudo-resolvents and applies degenerate semigroup theory to the solvability of ADAEs.

Findings

Established a new index concept for ADAEs

Applied pseudo-resolvent approach to practical examples

Demonstrated the effectiveness of degenerate semigroup methods

Abstract

We study linear abstract differential-algebraic equations (ADAEs), and we introduce an index concept which is based on polynomial growth of a~pseudo-resolvent. Our approach to solvability analysis is based on degenerate semigroups. We apply our results to some examples such as distributed circuit elements, and a system obtained by heat-wave coupling.