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

TL;DR

This work reconciles diverse regularity and index notions for linear time-varying DAEs $E(t)x'+F(t)x=q(t)$ by introducing canonical characteristic values and a flow-subspace $S_{can}$ that unify multiple frameworks. It distinguishes two broad analysis paths—elimination/dissection-based reductions and derivative-array approaches—and proves a main equivalence theorem tying thirteen definitions of regularity. The results show that, for regular DAEs, the index $\mu$ and the characteristic triplets $(r,\theta_0,\dots,\theta_{\mu-2})$ are invariant across formalisms and can be retrieved from any of the approaches, enabling seamless translation to standard canonical forms (SCF). The paper thus provides a unified, rank-driven view of DAE regularity, clarifying when a DAE is solvable, how the dynamical degree of freedom $d$ is determined, and how canonical forms reveal the underlying flow. These insights facilitate harmonized theoretical analysis and could improve numerical treatment and initialization for linear and nonlinear DAEs.

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.

Figures (12)

• Figure 1: Solution of \ref{['solution_x2']} for $M>0$ and $M<0$
• Figure 2: Solutions for $x_2$ and $x_3$ computed with MATHEMATICA, Version 13 for $\alpha(t)=0\text{ for }t\in (-\infty,0]t^{4}\text{ for }t\in (0,\infty)$, $q_1=q_2=q_3=0$ in Example \ref{['e.4']}. The difficulty to plot the solution around $0$ is due to the singularity.
• Figure 3: Behavior of solutions for the DAE \ref{['ex.2']} from Example \ref{['e.exp']} for $\gamma>0$ (left) or $\gamma<0$ (right), critical points (red) and stationary solutions (blue).
• Figure 4: Solution of the DAE \ref{['ex.2']} from Example \ref{['e.exp']} for $\gamma=-1$ and initial value $x_{0,1}=0.98$ (left), as well as solution of \ref{['ex.2_pert']} for $\delta_1(t)=0$, $\delta_2(t)=t^2$ (right), both for $t \in \left[0,1\right]$.
• Figure 5: Solution of the DAE \ref{['ex.2']} from Example \ref{['e.exp']} for $\gamma=-1$ and initial value $x_{0,1}=0.98$ (left), as well as solution of \ref{['ex.2_pert']} for $\delta_1(t)=0$, $\delta_2(t)=0.7 \sin(t)$ (right), both for $t \in \left[0,2\pi\right]$.

Theorems & Definitions (136)

• Definition 2.1
• Definition 2.2
• Definition 4.1
• Definition 4.2
• Definition 4.3
• Definition 4.4
• Remark 4.5
• Remark 4.6
• Theorem 4.7
• proof