The common ground of DAE approaches. An overview of diverse DAE frameworks emphasizing their commonalities

TL;DR

This paper provides a comprehensive overview of differential-algebraic equation (DAE) frameworks, identifying their commonalities through a unified regularity concept and canonical characteristic values, facilitating better understanding and analysis.

Contribution

It introduces a unified regularity framework for diverse DAE approaches, proving the equivalence of multiple definitions and emphasizing the importance of canonical characteristic values.

Findings

Proves the equivalence of thirteen regularity definitions.

Develops a comprehensive regularity concept applicable across frameworks.

Highlights the significance of canonical characteristic values in DAE analysis.

Abstract

We analyze different approaches to differential-algebraic equations with attention to the implemented rank conditions of various matrix functions. These conditions are apparently very different and certain rank drops in some matrix functions actually indicate a critical solution behavior. We look for common ground by considering various index and regularity notions from literature generalizing the Kronecker index of regular matrix pencils. In detail, starting from the most transparent reduction framework, we work out a comprehensive regularity concept with canonical characteristic values applicable across all frameworks and prove the equivalence of thirteen distinct definitions of regularity. This makes it possible to use the findings of all these concepts together. Additionally, we show why not only the index but also these canonical characteristic values are crucial to describe the properties of the DAE.