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Controllability of Fractional Linear Systems with Delays in Control

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Fractional Dynamical Systems: Methods, Algorithms and Applications

Part of the book series: Studies in Systems, Decision and Control ((SSDC,volume 402))

Abstract

In the chapter linear, fractional, continuous time, finite-dimensional, dynamical control systems with multiple variable point delays and distributed delay in admissible control described by linear ordinary differential state equations are considered. Using notations, theorems and methods taken directly from functional analysis and linear controllability theory, necessary and sufficient conditions for global relative controllability in a given finite time interval are formulated and proved. The main result of the chapter is to show, that global relative controllability of fractional linear systems with different types of delays in admissible control is equivalent to non-singularity of a suitably defined relative controllability matrix. In the proofs of the main results, methods and concepts taken from the theory of linear bounded operators in Hilbert spaces are used. Applying a relative controllability matrix for relative controllable systems steering admissible control is proposed, which steers the fractional system from a given initial complete state to the desired final relative state. Some remarks and comments on the existing controllability results for linear fractional dynamical system with delays are also presented.

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Acknowledgements

The research was founded by Polish National Research Centre under grant “The use of fractional order controllers in congestion control mechanism of Internet”, grant number UMO-2017/27/B/ST6/00145.

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Authors and Affiliations

  1. Institute of Theoretical and Applied Informatics, Polish Academy of Sciences, Bałtycka 5, 44-100, Gliwice, Poland

    Jerzy Klamka

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  1. Jerzy Klamka
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Correspondence to Jerzy Klamka .

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Editors and Affiliations

  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. (2022). Controllability of Fractional Linear Systems with Delays in Control. In: Kulczycki, P., Korbicz, J., Kacprzyk, J. (eds) Fractional Dynamical Systems: Methods, Algorithms and Applications. Studies in Systems, Decision and Control, vol 402. Springer, Cham. https://doi.org/10.1007/978-3-030-89972-1_11

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