Stability Analysis of Systems With Time-Varying Delays via an Improved Integral Inequality

Tian, Junkang; Ren, Zerong

doi:10.1109/access.2020.2994510

Cited by 4 publications

(14 citation statements)

References 24 publications

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“…The interest of applied mathematicians, engineers, etc., to investigate qualitative properties of solutions for such numerous problems with time-varying delays has increased considerably in the last decades. In particular, see the mentioned books, the papers of Arino et al [8], Azbelev et al [9], Berezansky and Braverman [10], Du [11], Gil [12], Graef and Tunç [13], Slyn'ko and Tunç [14], Tian and Ren [15], Tunç [16][17][18], Tunç and Erdur [19], Tunç and Golmankhaneh [20], Tunç and Tunç [21][22][23], Zevin [24] and bibliographies therein.…”

Section: Introductionmentioning

confidence: 99%

“…The motivation of this paper was inspired by a recent work of Tian and Ren [15]. From this point of view, we mention a related work of Tian and Ren [15].…”

Section: Introductionmentioning

confidence: 99%

“…The motivation of this paper was inspired by a recent work of Tian and Ren [15]. From this point of view, we mention a related work of Tian and Ren [15]. In 2020, Tian and Ren [15] considered the following system of linear DDEs with time-varying delay,…”

Section: Introductionmentioning

confidence: 99%

“…Tian and Ren [15] defined an LKF for the system of DDEs (1). Based on that LKF, Tian and Ren [15] proved a theorem, ( [15], Theorem 1), on the asymptotically stability of the system of DDEs (1).…”

Section: Introductionmentioning

confidence: 99%

“…(1) We investigate the uniformly asymptotically stability of zero solution of the system of DDEs (1), see Theorem 3. To investigate this problem, we define a very different LKF from that in Tian and Ren [15]. (2) We study the uniformly asymptotically stability of zero solution and the integrability of the norm of solutions of the following unperturbed nonlinear system of DDEs by Theorem 4 and Theorem 5, respectively:…”

Section: Introductionmentioning

confidence: 99%

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Qualitative Analyses of Differential Systems with Time-Varying Delays via Lyapunov–Krasovskiĭ Approach

Tunç

Wang

et al. 2021

Mathematics

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In this paper, a class of systems of linear and non-linear delay differential equations (DDEs) of first order with time-varying delay is considered. We obtain new sufficient conditions for uniform asymptotic stability of zero solution, integrability of solutions of an unperturbed system and boundedness of solutions of a perturbed system. We construct two appropriate Lyapunov–Krasovskiĭ functionals (LKFs) as the main tools in proofs. The technique of the proofs depends upon the Lyapunov–Krasovskiĭ method. For illustration, two examples are provided in particular cases. An advantage of the new LKFs used here is that they allow to eliminate using Gronwall’s inequality. When we compare our results with recent results in the literature, the established conditions are more general, less restrictive and optimal for applications.

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Section: Introductionmentioning

confidence: 99%

“…The motivation of this paper was inspired by a recent work of Tian and Ren [15]. From this point of view, we mention a related work of Tian and Ren [15].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Qualitative Analyses of Differential Systems with Time-Varying Delays via Lyapunov–Krasovskiĭ Approach

Tunç

Wang

et al. 2021

Mathematics

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Stability tests and solution estimates for non-linear differential equations

Tunç

2023

Int. J. Optim. Control, Theor. Appl. (IJOCTA)

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This article deals with certain systems of delay differential equations (DDEs) and a system of ordinary differential equations (ODEs). Here, five new theorems are proved on the fundamental properties of solutions of these systems. The results on the properties of solutions consist of sufficient conditions and they dealt with uniformly asymptotically stability (UAS), instability and integrability of solutions of unperturbed systems of DDEs, boundedness of solutions of a perturbed system of DDEs at infinity and exponentially stability (ES) of solutions of a system of nonlinear ODEs. Here, the techniques of proofs depend upon the Lyapunov- Krasovski? functional (LKF) method and Lyapunov function (LF) method. For illustrations, in particular cases, four examples are constructed as applications. Some results of this paper are given at first time in the literature, and the other results generalize and improve some related ones in the literature.

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Improved New Conditions for Qualitative Behaviors of Time-varying Delay Differential Systems

Tunç

2024

Muş Alparslan Üniversitesi Fen Bilimleri Dergisi

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Bilindiği üzere uygulamalı bilim alanlarında, öğreneğin, mühendislik, tıp, ekonomi, yapay sinir ağları vb. bir çok uygulamalı alanda ortaya çıkan problemlerin matematiksel olarak modellenmesine diferansiyel denklemler karşılık gelmektedir. Bu diferansiyel denklemlerin önemli bir türü is gecikmeli diferansiyel denklemlerdir. Lyapunov anlamında bu tür denklemlerin çözümlerinin karalılık vb. problemleri uygulamalarda önemli bir yere sahiptir. Ancak, genel anlamda gecikmeli diferansiyel denklemleri çözmek pek de kolay değildir, hatta numerik olarak hariç, genel anlamda açıkça çözümleri bulmak imkânsızdır. Bu tür zorluklara rağmen, Lyapunov-Krasovskii fonksiyonel metodu, göz önüne alınan gecikmeli diferansiyel denklemleri çözmeksizin, yani çözümler hakkında herhangi bir ön bilgiye ihtiyaç duymaksızın, çözümlerin kararlılığı vb. davranışları hakkında bilgi edinilmesine olanak tanır. Bu makalede, birinci mertebeden gecikmeli diferansiyel denklemlerin sürekli zaman sistemlerinin çözümlerinin bazı niteliksel analizleri ele alınmaktadır. Burada belli formda sürekli zaman gecikmeli pertürbe ve pertürbe olmayan diferansiyel denklemler sistemleri sırasıyla göz önüne alınmaktadır. Bu sistemlerin çözümlerinin asimptotik kararlılık, integrallenebilirlik ve sınırlılık davranışları Lyapunov-Krasovskii fonksiyonel metodu yardımıyla incelenmektedir. Elden edilen sonuçların uygulanabilirliğini göstermek için iki örnek verilmiştir. Verilen yeni sonuçlar, geçmiş literatürde elde edilmiş sonuçlardan daha genel niteliktedir

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Stability Analysis of Systems With Time-Varying Delays via an Improved Integral Inequality

Cited by 4 publications

References 24 publications

Qualitative Analyses of Differential Systems with Time-Varying Delays via Lyapunov–Krasovskiĭ Approach

Qualitative Analyses of Differential Systems with Time-Varying Delays via Lyapunov–Krasovskiĭ Approach

Stability tests and solution estimates for non-linear differential equations

Improved New Conditions for Qualitative Behaviors of Time-varying Delay Differential Systems

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