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Boundary disturbance rejection for fractional distributed parameter systems via the sliding mode and Riesz basis approach

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Abstract

We study the sliding mode control (SMC) design for unstable fractional heat and wave equations involving unknown external disturbances, respectively. In the case that the disturbance vanishes, a backstepping transform is constructed at first to stabilize the considered system in Mittag-Leffler (M-L) sense. When the disturbance flows into the boundary, sliding mode controllers are provided and the reaching conditions are also verified for fractional heat and wave systems, respectively. In the light of the Riesz basis approach, the well-posedness and closed-loop algebraic stability conclusions are established for fractional partial differential inclusion systems with discontinuous boundary conditions. As a by-product, a longtime unsolved problems raised in [Nonlinear Dynam, 38 (2004), 339-354], which is the first contribution about the boundary feedback stabilization control for fractional partial differential equations (PDEs), are completely solved with rigorous proof.

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Acknowledgements

This work is supported by the Natural Science Foundation of Shanghai (No. 19ZR1400500), the National Natural Science Foundation of China (No. 62173348, 12161141013, 62273239) and Hunan Provincial Natural Science Foundation of China (No. 2021JJ20081).

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Authors and Affiliations

  1. College of Science, University of Shanghai for Science and Technology, Shanghai, 200093, China

    Rui-Yang Cai

  2. School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha, 410075, China

    Hua-Cheng Zhou

  3. Department of Applied Mathematics, Donghua University, Shanghai, 201620, China

    Chun-Hai Kou

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  1. Rui-Yang Cai
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Correspondence to Hua-Cheng Zhou.

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Cai, RY., Zhou, HC. & Kou, CH. Boundary disturbance rejection for fractional distributed parameter systems via the sliding mode and Riesz basis approach. Nonlinear Dyn 111, 1355–1367 (2023). https://doi.org/10.1007/s11071-022-07897-3

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  • DOI: https://doi.org/10.1007/s11071-022-07897-3

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