Parameterization of Seed Functions for Equivalent Representations of Time-Varying Delay Systems
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866910055243186176 |
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| author | Kisole, Sengiyumva Chun, Jungbae Seiler, Peter Peet, Matthew M. |
| author_facet | Kisole, Sengiyumva Chun, Jungbae Seiler, Peter Peet, Matthew M. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_15948 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Parameterization of Seed Functions for Equivalent Representations of Time-Varying Delay Systems Kisole, Sengiyumva Chun, Jungbae Seiler, Peter Peet, Matthew M. Optimization and Control Systems and Control 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. |
| title | Parameterization of Seed Functions for Equivalent Representations of Time-Varying Delay Systems |
| topic | Optimization and Control Systems and Control |
| url | https://arxiv.org/abs/2603.15948 |