Discontinuity propagation in delay differential-algebraic equations

TL;DR

This paper provides a comprehensive analysis of how primary discontinuities propagate in linear delay differential-algebraic equations, introducing a new classification based on matrix pencil forms and considering all possible inhomogeneities.

Contribution

It offers a complete characterization of discontinuity propagation in DDAEs using (quasi-) Weierstra{ss} form and develops algebraic criteria for classification, including hidden delays.

Findings

Complete classification of propagation types

Algebraic criteria for analyzing DDAEs

Identification of hidden delays in DDAEs

Abstract

The propagation of primary discontinuities in initial value problems for linear delay differential-algebraic equations (DDAEs) is discussed. Based on the (quasi-) Weierstra{\ss} form for regular matrix pencil, a complete characterization of the different propagation types is given and algebraic criteria in terms of the matrices are developed. The analysis, which is based on the method of steps, takes into account all possible inhomogeneities and history functions and thus serves as the worst-case scenario. Moreover, it reveals possible hidden delays in the DDAE. The new classification for DDAEs is compared to existing approaches in the literature and the impact of splicing conditions on the classification is studied.