Control of Oscillating Systems with a Single Delay

Diblı́k, Josef; Khusainov, D. Ya.; Lukáčová, J; Růžičková, Miroslava

doi:10.1186/1687-1847-2010-108218

Cited by 19 publications

(5 citation statements)

References 2 publications

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“…In recent decades, the controllability of differential delay systems has been studied by many authors. There are a few recent studies in the literature on control theory [19][20][21][22][23][24] and Ulam stability [25][26][27][28] for delay differential equations.…”

Section: Introductionmentioning

confidence: 99%

Controllability and Hyers–Ulam Stability of Differential Systems with Pure Delay

Elshenhab

Wang

2022

Mathematics

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Dynamic systems of linear and nonlinear differential equations with pure delay are considered in this study. As an application, the representation of solutions of these systems with the help of their delayed Mittag–Leffler matrix functions is used to obtain the controllability and Hyers–Ulam stability results. By introducing a delay Gramian matrix, we establish some sufficient and necessary conditions for the controllability of linear delay differential systems. In addition, by applying Krasnoselskii’s fixed point theorem, we establish some sufficient conditions of controllability and Hyers–Ulam stability of nonlinear delay differential systems. Our results improve, extend, and complement some existing ones. Finally, two examples are given to illustrate the main results.

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Section: Introductionmentioning

confidence: 99%

Controllability and Hyers–Ulam Stability of Differential Systems with Pure Delay

Elshenhab

Wang

2022

Mathematics

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“…where cos τ Ω : R → R n×n and sin τ Ω : R → R n×n denote the delayed matrix cosine and the delayed matrix sine, respectively. It should be emphasized that the pioneer works [1][2][3][4][5][6][7][8][9][10][11] led to many new results in differential equations with a delay of integer and non-integer order and a discrete system with delay; see . These models have applications in oscillatory systems [24,25], computational mathematics [26], spatially extended fractional reaction-diffusion models [27], and so on.…”

Section: Introductionmentioning

confidence: 99%

Relative Controllability and Ulam–Hyers Stability of the Second-Order Linear Time-Delay Systems

2023

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We introduce the delayed sine/cosine-type matrix function and use the Laplace transform method to obtain a closed form solution to IVP for a second-order time-delayed linear system with noncommutative matrices A and Ω. We also introduce a delay Gramian matrix and examine a relative controllability linear/semi-linear time delay system. We have obtained the necessary and sufficient condition for the relative controllability of the linear time-delayed second-order system. In addition, we have obtained sufficient conditions for the relative controllability of the semi-linear second-order time-delay system. Finally, we investigate the Ulam–Hyers stability of a second-order semi-linear time-delayed system.

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“…Khusainov and Diblík [4] transferred this idea for solving the Cauchy problem for an oscillating system with second order and pure delay, by constructing special delayed matrix of cosine and sine type. These pioneer works led to many new results in integer and fractional order differential equations with delays and discrete delayed system; see [1,2,3,6,7,8,9,10,11,12,13,14,15,16,17,18,24,25].…”

Section: Introductionmentioning

confidence: 99%

Representation of solutions of a second order delay differential equation

Qiu,

Wang

2020

ejde

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In this article, we study an inhomogeneous second order delay differential equation on the fractal set \(\mathbb{R}^{\alpha n}\) \((0<\alpha\leq 1)\), based on the theory of local calculus. We introduce delay cosine and sine type matrix functions and give their properties on the fractal set. We give the representation of solutions to second order differential equations with pure delay and two delays. For more information see https://ejde.math.txstate.edu/Volumes/2020/72/abstr.html

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Control of Oscillating Systems with a Single Delay

Cited by 19 publications

References 2 publications

Controllability and Hyers–Ulam Stability of Differential Systems with Pure Delay

Controllability and Hyers–Ulam Stability of Differential Systems with Pure Delay

Relative Controllability and Ulam–Hyers Stability of the Second-Order Linear Time-Delay Systems

Representation of solutions of a second order delay differential equation

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