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Approximate Controllability of Infinite-Dimensional Degenerate Fractional Order Systems in the Sectorial Case

Author

Listed:
  • Dumitru Baleanu

    (Department of Mathematics, Faculty of Arts and Sciences, Çankaya University, TR-06530 Ankara, Turkey
    Institute of Space Science, R-077125 Măgurle-Bucharest, Romania)

  • Vladimir E. Fedorov

    (Department of Mathematical Analysis, Chelyabinsk State University, 454001 Chelyabinsk, Russia
    Laboratory of Functional Materials, South Ural State University, 454080 Chelaybinsk, Russia)

  • Dmitriy M. Gordievskikh

    (Department of Physics, Mathematics and Information Technology Education, Shadrinsk State Pedagogical University, 641870 Shadrinsk, Russia)

  • Kenan Taş

    (Department of Mathematics, Faculty of Arts and Sciences, Çankaya University, TR-06530 Ankara, Turkey)

Abstract

We consider a class of linear inhomogeneous equations in a Banach space not solvable with respect to the fractional Caputo derivative. Such equations are called degenerate. We study the case of the existence of a resolving operators family for the respective homogeneous equation, which is an analytic in a sector. The existence of a unique solution of the Cauchy problem and of the Showalter—Sidorov problem to the inhomogeneous degenerate equation is proved. We also derive the form of the solution. The approximate controllability of infinite-dimensional control systems, described by the equations of the considered class, is researched. An approximate controllability criterion for the degenerate fractional order control system is obtained. The criterion is illustrated by the application to a system, which is described by an initial-boundary value problem for a partial differential equation, not solvable with respect to the time-fractional derivative. As a corollary of general results, an approximate controllability criterion is obtained for the degenerate fractional order control system with a finite-dimensional input.

Suggested Citation

  • Dumitru Baleanu & Vladimir E. Fedorov & Dmitriy M. Gordievskikh & Kenan Taş, 2019. "Approximate Controllability of Infinite-Dimensional Degenerate Fractional Order Systems in the Sectorial Case," Mathematics, MDPI, vol. 7(8), pages 1-15, August.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:8:p:735-:d:256864
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    Citations

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    Cited by:

    1. Marina Plekhanova & Guzel Baybulatova, 2020. "Multi-Term Fractional Degenerate Evolution Equations and Optimal Control Problems," Mathematics, MDPI, vol. 8(4), pages 1-9, April.
    2. Jiae Yang & Yujia Wang & Tong Wang & Xuebo Yang, 2022. "Fuzzy-Based Tracking Control for a Class of Fractional-Order Systems with Time Delays," Mathematics, MDPI, vol. 10(11), pages 1-22, May.
    3. Amar Debbouche & Vladimir E. Fedorov, 2020. "A Class of Fractional Degenerate Evolution Equations with Delay," Mathematics, MDPI, vol. 8(10), pages 1-8, October.

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