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arXiv · 2603.15948

Parameterization of Seed Functions for Equivalent Representations of Time-Varying Delay Systems

Sengiyumva Kisole·Jungbae Chun·Peter Seiler·Matthew M. Peet
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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 $L_2$. 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.

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Sengiyumva Kisole, Jungbae Chun, Peter Seiler, Matthew M. Peet. 2026-03-16. Parameterization of Seed Functions for Equivalent Representations of Time-Varying Delay Systems. https://arxiv.org/abs/2603.15948

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