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

Author

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  • Ahmed M. Elshenhab

    (School of Mathematics, Harbin Institute of Technology, Harbin 150001, China
    These authors contributed equally to this work.)

  • Xingtao Wang

    (School of Mathematics, Harbin Institute of Technology, Harbin 150001, China
    These authors contributed equally to this work.)

Abstract

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.

Suggested Citation

  • Ahmed M. Elshenhab & Xingtao Wang, 2022. "Controllability and Hyers–Ulam Stability of Differential Systems with Pure Delay," Mathematics, MDPI, vol. 10(8), pages 1-18, April.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:8:p:1248-:d:790916
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    References listed on IDEAS

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    1. Elshenhab, Ahmed M. & Wang, Xing Tao, 2021. "Representation of solutions of linear differential systems with pure delay and multiple delays with linear parts given by non-permutable matrices," Applied Mathematics and Computation, Elsevier, vol. 410(C).
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    Cited by:

    1. Huang, Jizhao & Luo, Danfeng & Zhu, Quanxin, 2023. "Relatively exact controllability for fractional stochastic delay differential equations of order κ∈(1,2]," Chaos, Solitons & Fractals, Elsevier, vol. 170(C).

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