Adaptive error feedback tracking for a wave equation with a nonlocal term and multi-channel unknown harmonic disturbances

Xinting Xiao; Feng-Fei Jin; Xiyu Liu; Xinting Xiao; Feng-Fei Jin; Xiyu Liu

doi:10.3934/math.2026012

AIMS Mathematics

2026, Volume 11, Issue 1: 291-321. doi: 10.3934/math.2026012

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Research article Topical Sections

Adaptive error feedback tracking for a wave equation with a nonlocal term and multi-channel unknown harmonic disturbances

1.
School of Mathematics and Statistics, Shandong Normal University, Jinan 250358, China
2.
Business School, Shandong Normal University, Jinan 250358, China

Received: 30 October 2025 Revised: 10 December 2025 Accepted: 24 December 2025 Published: 05 January 2026
MSC : 93C20

This paper addressed the output regulation problem for a one-dimensional (1-D) wave equation subject to a nonlocal term and multi-channel unknown disturbances. Motivated by the combined challenge of structural instability from nonlocal coupling and the realistic presence of multi-channel harmonic disturbances with unknown frequencies, this work aimed to integrate and simultaneously address both issues to meet more complex application scenarios. The nonlocal term caused energy growth and open-loop instability, requiring sequential stabilization and output regulation. The disturbances consisted of sinusoidal signals with unknown amplitudes and frequencies, where only an upper bound on the frequencies was known. Our approach constructed an auxiliary system to eliminate the nonlocal effect and employed a coordinate transformation that concentrated disturbances into the tracking error channel. An adaptive observer was then developed for online frequency identification, enabling output-feedback control using the tracking error and its derivative. Theoretical analysis established the well-posedness and state boundedness of the closed-loop system, while numerical simulations confirmed the effectiveness of the proposed approach and demonstrated exponential convergence of the tracking error to zero.
- wave equation,
- output regulation,
- adaptive observer,
- disturbance rejection
Citation: Xinting Xiao, Feng-Fei Jin, Xiyu Liu. Adaptive error feedback tracking for a wave equation with a nonlocal term and multi-channel unknown harmonic disturbances[J]. AIMS Mathematics, 2026, 11(1): 291-321. doi: 10.3934/math.2026012

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Abstract

This paper addressed the output regulation problem for a one-dimensional (1-D) wave equation subject to a nonlocal term and multi-channel unknown disturbances. Motivated by the combined challenge of structural instability from nonlocal coupling and the realistic presence of multi-channel harmonic disturbances with unknown frequencies, this work aimed to integrate and simultaneously address both issues to meet more complex application scenarios. The nonlocal term caused energy growth and open-loop instability, requiring sequential stabilization and output regulation. The disturbances consisted of sinusoidal signals with unknown amplitudes and frequencies, where only an upper bound on the frequencies was known. Our approach constructed an auxiliary system to eliminate the nonlocal effect and employed a coordinate transformation that concentrated disturbances into the tracking error channel. An adaptive observer was then developed for online frequency identification, enabling output-feedback control using the tracking error and its derivative. Theoretical analysis established the well-posedness and state boundedness of the closed-loop system, while numerical simulations confirmed the effectiveness of the proposed approach and demonstrated exponential convergence of the tracking error to zero.

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