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Controllability and Stability of Semilinear Fractional Order Systems

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Automatic Control, Robotics, and Information Processing

Part of the book series: Studies in Systems, Decision and Control ((SSDC,volume 296))

Abstract

In the chapter two the most important properties of fractional order dynamical systems, namely, controllability and stability are presented. At the beginning the basic notations and the fundamental definitions are recalled. The first part of the chapter is devoted to controllability and contains the formulation of the problem, main hypotheses and theorems about controllability of semilinear fractional order systems with distributed and point multiplicities constant or variable delays in the state variables and controls. Next, using fixed point theorems approximate controllability problem in infinite dimensional spaces, in particular Banach or Hilbert space, are discussed. Second part of the chapter is devoted to the stability problem of fractional order systems. The problem of stability and the problem of the existence of solutions for linear and nonlinear fractional order systems are also presented.

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Acknowledgments

The research presented here was done as parts of the projects funded by the National Science Centre in Poland granted according to decisions UMO-2017/25/B/ST7/ 02236 (JK) and DEC-2017/25/B/ST7/02888 (AB, MN). The work of Adam Czornik was supported by Polish Ministry for Science and Higher Education funding for statutory activities 02/990/BK_19/0121.

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Authors and Affiliations

  1. Silesian University of Technology, Faculty of Automatic Control, Electronics and Computer Science, Akademicka 16, 44-100, Gliwice, Poland

    Jerzy Klamka, Artur Babiarz, Adam Czornik & Michał Niezabitowski

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  1. Jerzy Klamka
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  2. Artur Babiarz
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  3. Adam Czornik
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  4. Michał Niezabitowski
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Correspondence to Jerzy Klamka .

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  1. Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

    Piotr Kulczycki

  2. Institute of Control and Computation Engineering, University of Zielona Góra, Zielona Góra, Poland

    Józef Korbicz

  3. Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

    Janusz Kacprzyk

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Klamka, J., Babiarz, A., Czornik, A., Niezabitowski, M. (2021). Controllability and Stability of Semilinear Fractional Order Systems. In: Kulczycki, P., Korbicz, J., Kacprzyk, J. (eds) Automatic Control, Robotics, and Information Processing. Studies in Systems, Decision and Control, vol 296. Springer, Cham. https://doi.org/10.1007/978-3-030-48587-0_9

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