Unimodular transformations for DAE initial trajectory problems

In order to compare solutions of (3) and (4) we make the following definition. Definition 2.1 The DAE (2), respectively the ITP (3), is called regular if, and only if, for all initial trajectories x ∈ DpwC∞ and all inhomogeneities f ∈ DpwC∞ a unique solution x ∈ DpwC∞ for the ITP (3) exists. The ITP (3) can be interpreted as a switched DAE with a switch at time t = 0. Following [1, Thm. 3.5.2] and the proof of [2, Thm. 7] we conclude that the ITP (3) is regular if, and only if, ` = n and det(P (s)) 6≡ 0. Unimodularity of U(s) and V (s) implies that the ITP (3) is regular if, and only if, the transformed ITP (4) is regular. Before we present our main result, we need the following technical observation. Lemma 2.2 Consider F ∈ DpwC∞ and U(s) = ∑d k=0 Uks k ∈ R[s]`×`. Then