Parameterization of Seed Functions for Equivalent Representations of Time-Varying Delay Systems
Abstract
Abel's classic transformation shows that any well-posed system with time-varying delay is equivalent to a parameter-varying system with fixed delay. The existence of such a parameter-varying constant delay representation then simplifies the problems of stability analysis and optimal control. Unfortunately, the method for construction of such transformations has been ad-hoc -- requiring an iterative time-stepping approach to constructing the transformation beginning with a seed function subject to boundary-value constraints. Moreover, a poor choice of seed function often results in a constant delay representation with large time-variations in system parameters -- obviating the benefits of such a representation. In this paper, we show how the set of all feasible seed functions can be parameterized using a basis for . This parameterization is then used to search for seed functions for which the corresponding time-transformation results in smaller parameter variation. The parameterization of admissible seed functions is illustrated with numerical examples that contrast how well-chosen and poorly chosen seed functions affect the boundedness of a time transformation.
Cite
@article{arxiv.2603.15948,
title = {Parameterization of Seed Functions for Equivalent Representations of Time-Varying Delay Systems},
author = {Sengiyumva Kisole and Jungbae Chun and Peter Seiler and Matthew M. Peet},
journal= {arXiv preprint arXiv:2603.15948},
year = {2026}
}