Asymptotic stability of solutions to a class of linear time-delay systems with periodic coefficients and a large parameter
- Gennadii V Demidenko^{1, 2}Email author and
- Inessa I Matveeva^{1, 2}
https://doi.org/10.1186/s13660-015-0853-7
© Demidenko and Matveeva 2015
Received: 13 April 2015
Accepted: 25 September 2015
Published: 13 October 2015
Abstract
A class of linear time-delay systems with periodic coefficients and a large parameter is studied. We establish conditions under which the zero solution is asymptotically stable. This result allows us to study the asymptotic stability of the zero solution to the time-delay systems without spectral methods and Lyapunov-Krasovskii functionals.
Keywords
MSC
1 Introduction
There are a large number of works devoted to delay differential equations (for instance, see [1–11] and the bibliography therein). One of the interesting questions is the asymptotic stability of solutions to delay differential equations. This question is very important from theoretical and practical viewpoints, because delay differential equations arise in many applied problems when describing the processes whose rates of change are defined by present and previous states (for example, see [12–14] and the bibliography therein). The case of variable coefficients is of special interest and is more complicated in comparison with the case of constant coefficients.
2 Preliminaries
In [15–20] we studied the question about the asymptotic stability of solutions to systems of ordinary differential equations and systems of delay differential equations with periodic coefficients. We proved theorems on asymptotic stability which are analogs of the classic theorems on the asymptotic stability for equations with constant coefficients. We now formulate two theorems used hereinafter.
Theorem 1
Theorem 2
Using the mentioned matrices \(L(t)\), \(K(s)\), estimates characterizing exponential decay of solutions to (5) at infinity were established in [16, 17].
Theorem 3
Proof
The theorem is proved. □
Remark 1
It should be noted that N satisfying the conditions of Theorem 3 exists owing to the continuity of the entries of \(A(t)\) on \([0, T]\).
Remark 2
It follows from the proof that all solutions to (9) tend to zero as \(t \to\infty\).
3 Asymptotic stability of solutions to delay differential equations
We now consider the time-delay systems of the form (1). Using the theorems formulated in Section 2, we indicate a value \(\mu_{\ast}\) such that the zero solution to (1) is asymptotically stable for \(\mu> \mu_{\ast}\).
Theorem 4
Proof
Theorem 4 is proved. □
Remark 3
It follows from the proof that all solutions to (1) tend to zero as \(t \to\infty\).
4 Conclusion
In the present paper we considered the systems of delay differential equations with periodic coefficients of the form (1). We established the conditions on the coefficients and the parameter under which the zero solution is asymptotically stable. These conditions are formulated in terms of inequalities. This result allows us to study the asymptotic stability of the zero solution to time-delay systems of such type without spectral methods (analogs of the Lyapunov-Floquet theory) and Lyapunov-Krasovskii functionals. In this connection let us note [24, 25] that the stability of solutions to nonautonomous linear delay differential systems has been studied by using the approaches based on the so-called Azbelev W-transform and the Bohl-Perron type theorem, respectively. Using the results of [16, 17], it is easy to write down the estimates for solutions to (1) characterizing the exponential decay rate at infinity.
Declarations
Acknowledgements
The authors were supported by the Russian Foundation for Basic Research (project no. 13-01-00329). The authors are grateful to the anonymous referees for the helpful comments and suggestions.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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