Regular linear time varying DAEs are equivalent to DAEs in strong standard canonical form

TL;DR

This work tackles when a linear DAE in standard form $E x'+F x=q$ with rank constraints is transformable to a strong standard canonical form (SSCF) featuring a constant nilpotent $N$. Building on prior equivalence frameworks and block-structured canonical forms, the authors develop two constructive procedures under full-column and full-row rank assumptions, respectively, using smooth singular value decompositions and structured similarity transforms to realize a constant $N$. A key contribution is the establishment of a fourteenth equivalent characterization of regularity via transformability to SSCF, with the Jordan structure of $N$ directly encoding the canonical characteristics $(\mu,\theta_i,d)$. The results yield a finite, algorithmic path to SSCF from block-structured canonical forms, enabling stable numerical solution and clearer index interpretation for regular linear DAEs.

Abstract

The relationship between solvability of linear diffential-algebraic equations (DAEs) and their transformability into canonical forms has been investigated for more than forty years. After a comparative analysis of numerous DAE frameworks the notions regularity and almost regularity were established only recently. Regular DAEs resulted to be equivalently transformable into so-called standard canonical forms (SCF) with block-structured nilpotent matrix functions featuring certain rank properties. In this paper we prove that for regular DAEs, even a transformation into a strong standard canonical form (SSCF) is possible, i.e. a SCF with a constant nilpotent matrix. We start from block-structured SCF and give a constructive proof.

Figures (9)

• Figure 1: Powers of $N$ with $N^5=0$ for $N$ consisting of $7$ Jordan blocks, $m=26$, $r=\operatorname{rank} N=18$, $d=0$, $\theta_0= 7, \theta_1= 5, \theta_2= 4, \theta_3=2$. This corresponds to two blocks of order 5, two blocks of order 4, one block of order 3, two blocks order 2 and one block order 1.
• Figure 2: Powers of $N$ with $N^5=0$ for $N$ having secondary blocks of full column rank, $m=26$, $r=18$, $\theta_0= 7, \theta_1= 5, \theta_2= 4, \theta_3=2$.
• Figure 3: Powers of $N$ with $N^5=0$ for $N$ having secondary blocks of full row rank, $m=26$, $r=18$, $\theta_0= 7, \theta_1= 5, \theta_2= 4, \theta_3=2$.
• Figure 4: Powers of $N=N^{(Ec)}$ with $N^5=0$ for $N$ with full column-rank secondary blocks, $m=26$, $r=18$, $\theta_0= 7, \theta_1= 5, \theta_2= 4, \theta_3=2$.
• Figure 5: Visualization of $N^{(k)}=L^{(k-1)}N^{(k-1)}K^{(k-1)}$ for $\mu$ steps starting from $N$ (left) and ending with $N^{(Ec)}$ for $N'\equiv 0$.

Theorems & Definitions (13)

• Conjecture 1.1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Proposition 4.1
• proof
• Lemma 4.2
• proof
• Proposition 5.1