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Controllability and Observability Results of an Implicit Type Fractional Order Delay Dynamical System

Author

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  • Irshad Ahmad

    (Department of Mathematics, University of Malakand, Chakdara 18800, Khyber Pakhtunkhwa, Pakistan)

  • Saeed Ahmad

    (Department of Mathematics, University of Malakand, Chakdara 18800, Khyber Pakhtunkhwa, Pakistan)

  • Ghaus ur Rahman

    (Department of Mathematics and Statistics, University of Swat, Swat 01923, Khyber Pakhtunkhwa, Pakistan)

  • Shabir Ahmad

    (Department of Mathematics, University of Malakand, Chakdara 18800, Khyber Pakhtunkhwa, Pakistan)

  • Manuel De la Sen

    (Department of Electricity and Electronics, Institute of Research and Development of Processes, Faculty of Science and Technology, University of the Basque Country, Campus of Leioa (Bizkaia), 48940 Leioa, Spain)

Abstract

Recently, several research articles have investigated the existence of solutions for dynamical systems with fractional order and their controllability. Nevertheless, very little attention has been given to the observability of such dynamical systems. In the present work, we explore the outcomes of controllability and observability regarding a differential system of fractional order with input delay. Laplace and inverse Laplace transforms, along with the Mittage–Leffler matrix function, are applied to the proposed dynamical system in Caputo’s sense, and a general solution is obtained in the form of an integral equation. Then, we set out conditions for the controllability of the underlying model, regarding the linear case. We then expound controllability conditions for the nonlinear case by utilizing the fixed point result of Schaefer and the Arzola–Ascoli theorem. Using the fixed point concept, we prove the observability of the linear case using the observability Grammian matrix. The necessary and sufficient conditions for the nonlinear case are investigated with the help of the Banach contraction mapping theorem. Finally, we add some examples to elaborate on our work.

Suggested Citation

  • Irshad Ahmad & Saeed Ahmad & Ghaus ur Rahman & Shabir Ahmad & Manuel De la Sen, 2022. "Controllability and Observability Results of an Implicit Type Fractional Order Delay Dynamical System," Mathematics, MDPI, vol. 10(23), pages 1-24, November.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:23:p:4466-:d:984949
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    References listed on IDEAS

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    1. Tamilalagan, P. & Balasubramaniam, P., 2017. "Moment stability via resolvent operators of fractional stochastic differential inclusions driven by fractional Brownian motion," Applied Mathematics and Computation, Elsevier, vol. 305(C), pages 299-307.
    2. Shi, Jianping & He, Ke & Fang, Hui, 2022. "Chaos, Hopf bifurcation and control of a fractional-order delay financial system," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 194(C), pages 348-364.
    3. K. Balachandran & V. Govindaraj & M. Rivero & J. A. Tenreiro Machado & J. J. Trujillo, 2013. "Observability of Nonlinear Fractional Dynamical Systems," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-7, June.
    4. Yunhong Xu & Huadong Wang & Nga Lay Hui & Efthymios G. Tsionas, 2021. "Prediction of Agricultural Water Consumption in 2 Regions of China Based on Fractional-Order Cumulative Discrete Grey Model," Journal of Mathematics, Hindawi, vol. 2021, pages 1-7, December.
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