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A full characterization of the critical stability curves/surfaces of linear systems with multiple delays.

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Title: A full characterization of the critical stability curves/surfaces of linear systems with multiple delays.
Authors: Liang, Song1,2 (AUTHOR) songliang@ahau.edu.cn, Wang, Zaihua3 (AUTHOR) zhwang@nuaa.edu.cn
Source: Meccanica. Aug2025, Vol. 60 Issue 8, p2437-2450. 14p.
Subjects: Stability theory, Time delay systems, Nonlinear equations, Hypersurfaces, Linear systems, Bifurcation theory, Numerical analysis
Abstract: The stability analysis of a linear system with the multiple delays as parameters in given intervals is not a new but hard topic in general, for which a key step is to find out all the critical stability curves/surfaces in the parameter space. In this paper, the critical stability condition is regarded as a complex equations depending nonlinearly on the delays, and it is solved in three parts: (1) The solvability of the nonlinear equation; (2) The representation of the solutions; 3) Numerical algorithms for finding the solutions. For the solvability, a necessary and sufficient condition in terms of a delay-independent inequality with clear geometrical meaning has been derived from the critical stability condition in the form of vector equation. For the representation, the critical delays in nested form are expressed explicitly in terms of a number of hypersurfaces, all the quantities have clear geometrical meaning. Based on the nested representation, two effective algorithms are proposed for finding the solutions, and illustrated with simple examples. The main results not only generalize the previous ones for systems with two delays and three delays of the nondegenerate cases, but also add new findings for the degenerated cases which have important impact on the stability of the time-delay systems. [ABSTRACT FROM AUTHOR]
Copyright of Meccanica is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: A full characterization of the critical stability curves/surfaces of linear systems with multiple delays.
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  Data: <searchLink fieldCode="AR" term="%22Liang%2C+Song%22">Liang, Song</searchLink><relatesTo>1,2</relatesTo> (AUTHOR)<i> songliang@ahau.edu.cn</i><br /><searchLink fieldCode="AR" term="%22Wang%2C+Zaihua%22">Wang, Zaihua</searchLink><relatesTo>3</relatesTo> (AUTHOR)<i> zhwang@nuaa.edu.cn</i>
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  Data: <searchLink fieldCode="JN" term="%22Meccanica%22">Meccanica</searchLink>. Aug2025, Vol. 60 Issue 8, p2437-2450. 14p.
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  Data: <searchLink fieldCode="DE" term="%22Stability+theory%22">Stability theory</searchLink><br /><searchLink fieldCode="DE" term="%22Time+delay+systems%22">Time delay systems</searchLink><br /><searchLink fieldCode="DE" term="%22Nonlinear+equations%22">Nonlinear equations</searchLink><br /><searchLink fieldCode="DE" term="%22Hypersurfaces%22">Hypersurfaces</searchLink><br /><searchLink fieldCode="DE" term="%22Linear+systems%22">Linear systems</searchLink><br /><searchLink fieldCode="DE" term="%22Bifurcation+theory%22">Bifurcation theory</searchLink><br /><searchLink fieldCode="DE" term="%22Numerical+analysis%22">Numerical analysis</searchLink>
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: The stability analysis of a linear system with the multiple delays as parameters in given intervals is not a new but hard topic in general, for which a key step is to find out all the critical stability curves/surfaces in the parameter space. In this paper, the critical stability condition is regarded as a complex equations depending nonlinearly on the delays, and it is solved in three parts: (1) The solvability of the nonlinear equation; (2) The representation of the solutions; 3) Numerical algorithms for finding the solutions. For the solvability, a necessary and sufficient condition in terms of a delay-independent inequality with clear geometrical meaning has been derived from the critical stability condition in the form of vector equation. For the representation, the critical delays in nested form are expressed explicitly in terms of a number of hypersurfaces, all the quantities have clear geometrical meaning. Based on the nested representation, two effective algorithms are proposed for finding the solutions, and illustrated with simple examples. The main results not only generalize the previous ones for systems with two delays and three delays of the nondegenerate cases, but also add new findings for the degenerated cases which have important impact on the stability of the time-delay systems. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of Meccanica is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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      – Type: doi
        Value: 10.1007/s11012-025-02035-w
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      – Code: eng
        Text: English
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        PageCount: 14
        StartPage: 2437
    Subjects:
      – SubjectFull: Stability theory
        Type: general
      – SubjectFull: Time delay systems
        Type: general
      – SubjectFull: Nonlinear equations
        Type: general
      – SubjectFull: Hypersurfaces
        Type: general
      – SubjectFull: Linear systems
        Type: general
      – SubjectFull: Bifurcation theory
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      – SubjectFull: Numerical analysis
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            – D: 01
              M: 08
              Text: Aug2025
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              Y: 2025
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