The common ground of DAE approaches. An overview of diverse DAE frameworks emphasizing their commonalities
This work provides a foundational unification for researchers in numerical analysis and applied mathematics dealing with DAEs, enabling the integration of findings across different frameworks.
The paper tackles the problem of unifying diverse frameworks for differential-algebraic equations (DAEs) by analyzing rank conditions and matrix functions, resulting in a comprehensive regularity concept that proves the equivalence of thirteen distinct definitions of regularity and highlights the importance of canonical characteristic values beyond just the index.
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.