统一多种微分代数方程方法，揭示其核心共性与关键特征值
The common ground of DAE approaches. An overview of diverse DAE frameworks emphasizing their commonalities
- 提出通用正则性概念，涵盖十三种不同定义的等价性证明
- 发现指数与特征值共同决定DAE解的关键性质
- 适用于理论分析与多框架统一研究的学者
我们分析了微分代数方程（DAE）中不同方法对矩阵函数秩条件的实现，这些条件看似各异，但某些秩下降实际上标志着解行为的临界变化。通过考察文献中推广克罗内克指数的各类指标与正则性概念，从最清晰的化简框架出发，构建了一个跨所有框架适用的、包含标准特征值的综合性正则性概念，并证明了十三种不同正则性定义的等价性。这使得各类概念的研究成果可协同使用。此外，我们阐明了不仅指数，这些标准特征值也对描述DAE性质至关重要。
原文摘要 · Abstract (English)
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.
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