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Boundary control of a fractional reaction-diffusion equation coupled with fractional ordinary differential equations with delay

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  • Hou, Mimi
  • Xi, Xuan-Xuan
  • Zhou, Xian-Feng

Abstract

This paper deals with boundary feedback control for a fractional reaction-diffusion equation with varying coefficient coupled with fractional ordinary differential equations with delay, which is a generalization of integer order coupled system. By designing a state feedback controller, we transform an unstable system into an asymptotic stable system via the backstepping method. The exact solution of the target system is given by the Prabhakar function. We also obtain the exact solution of the original system with the help of the invertible coordinate transformation. Furthermore, by the fractional Halanay’s inequality, we structure a Lyapunov functional to prove the asymptotic stability of the given system. Finally, a numerical simulation example is provided to illustrate the applications of our results.

Suggested Citation

  • Hou, Mimi & Xi, Xuan-Xuan & Zhou, Xian-Feng, 2021. "Boundary control of a fractional reaction-diffusion equation coupled with fractional ordinary differential equations with delay," Applied Mathematics and Computation, Elsevier, vol. 406(C).
    • Handle: RePEc:eee:apmaco:v:406:y:2021:i:c:s0096300321003490
    DOI: 10.1016/j.amc.2021.126260
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    References listed on IDEAS

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    1. Torricelli, Lorenzo, 2020. "Trade duration risk in subdiffusive financial models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 541(C).
    2. Xian, Jun & Yan, Xiong-bin & Wei, Ting, 2020. "Simultaneous identification of three parameters in a time-fractional diffusion-wave equation by a part of boundary Cauchy data," Applied Mathematics and Computation, Elsevier, vol. 384(C).
    3. Zhou, Xian-Feng & Yang, Fuli & Jiang, Wei, 2015. "Analytic study on linear neutral fractional differential equations," Applied Mathematics and Computation, Elsevier, vol. 257(C), pages 295-307.
    4. Li, Hui & Kao, YongGui, 2019. "Mittag-Leffler stability for a new coupled system of fractional-order differential equations with impulses," Applied Mathematics and Computation, Elsevier, vol. 361(C), pages 22-31.
    5. Liu, Xiping & Jia, Mei, 2019. "Solvability and numerical simulations for BVPs of fractional coupled systems involving left and right fractional derivatives," Applied Mathematics and Computation, Elsevier, vol. 353(C), pages 230-242.
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    Cited by:

    1. Sun, Yuting & Hu, Cheng & Yu, Juan & Shi, Tingting, 2023. "Synchronization of fractional-order reaction-diffusion neural networks via mixed boundary control," Applied Mathematics and Computation, Elsevier, vol. 450(C).
    2. Zeng, Caijin & Zhou, Zhongcheng & Xie, Chengkang, 2024. "Null controllability of an ODE-heat system coupled at boundary and internal term," Applied Mathematics and Computation, Elsevier, vol. 475(C).
    3. Li, Xing-Yu & Wu, Kai-Ning & Liu, Xiao-Zhen, 2023. "Mittag–Leffler stabilization for short memory fractional reaction-diffusion systems via intermittent boundary control," Applied Mathematics and Computation, Elsevier, vol. 449(C).

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