Abstract

In this paper, a modification of the Krasovskii theorem for a nonlinear class of systems is presented. The idea is to replace the positive definiteness condition of the functional with this condition on the special Razumikhin-type set of functions only while retaining the other classical conditions. The result is motivated by the fact that this idea leads to the necessary and sufficient stability condition for linear time-invariant systems. Moreover, this condition is constructive and allows us not only to directly analyze the stability but also to find the robustness bounds on the matrix parameters and on the delays and to construct the exponential estimates for solutions. An overview of these results for linear systems is also presented.

Original languageEnglish
Pages (from-to)147-152
Number of pages6
JournalIFAC-PapersOnLine
Volume51
Issue number14
DOIs
Publication statusPublished - 1 Jan 2018

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Linear systems

Scopus subject areas

  • Control and Systems Engineering

Cite this

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title = "At the junction of Lyapunov-Krasovskii and Razumikhin approaches",
abstract = "In this paper, a modification of the Krasovskii theorem for a nonlinear class of systems is presented. The idea is to replace the positive definiteness condition of the functional with this condition on the special Razumikhin-type set of functions only while retaining the other classical conditions. The result is motivated by the fact that this idea leads to the necessary and sufficient stability condition for linear time-invariant systems. Moreover, this condition is constructive and allows us not only to directly analyze the stability but also to find the robustness bounds on the matrix parameters and on the delays and to construct the exponential estimates for solutions. An overview of these results for linear systems is also presented.",
keywords = "asymptotic stability, exponential stability, Lyapunov–Krasovskii functionals, Razumikhin condition, robust stability, time-delay systems, uncertain delay",
author = "Alexandrova, {Irina V.} and Zhabko, {Alexey P.}",
year = "2018",
month = "1",
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doi = "10.1016/j.ifacol.2018.07.214",
language = "English",
volume = "51",
pages = "147--152",
journal = "IFAC-PapersOnLine",
issn = "2405-8963",
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T1 - At the junction of Lyapunov-Krasovskii and Razumikhin approaches

AU - Alexandrova, Irina V.

AU - Zhabko, Alexey P.

PY - 2018/1/1

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N2 - In this paper, a modification of the Krasovskii theorem for a nonlinear class of systems is presented. The idea is to replace the positive definiteness condition of the functional with this condition on the special Razumikhin-type set of functions only while retaining the other classical conditions. The result is motivated by the fact that this idea leads to the necessary and sufficient stability condition for linear time-invariant systems. Moreover, this condition is constructive and allows us not only to directly analyze the stability but also to find the robustness bounds on the matrix parameters and on the delays and to construct the exponential estimates for solutions. An overview of these results for linear systems is also presented.

AB - In this paper, a modification of the Krasovskii theorem for a nonlinear class of systems is presented. The idea is to replace the positive definiteness condition of the functional with this condition on the special Razumikhin-type set of functions only while retaining the other classical conditions. The result is motivated by the fact that this idea leads to the necessary and sufficient stability condition for linear time-invariant systems. Moreover, this condition is constructive and allows us not only to directly analyze the stability but also to find the robustness bounds on the matrix parameters and on the delays and to construct the exponential estimates for solutions. An overview of these results for linear systems is also presented.

KW - asymptotic stability

KW - exponential stability

KW - Lyapunov–Krasovskii functionals

KW - Razumikhin condition

KW - robust stability

KW - time-delay systems

KW - uncertain delay

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