Stabilization and parameter estimation for 1-D wave equation with variable coefficients and uncertain harmonic disturbances

Yan-Na Jia; Xiu-Fang Yu; Can Jin

doi:10.3934/eect.2024067

Article Contents

2025, Volume 14, Issue 3: 530-541. Doi: 10.3934/eect.2024067

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Stabilization and parameter estimation for 1-D wave equation with variable coefficients and uncertain harmonic disturbances

1.
School of Mathematics and Statistics, Taiyuan Normal University, 030619 Taiyuan, Shanxi, China
2.
College of Mathematics, Taiyuan University of Technology, 030600 Taiyuan, Shanxi, China

^*Corresponding author: Xiu-Fang Yu
^*Corresponding author: Xiu-Fang Yu

Received: September 1, 2024

Revised: October 1, 2024

Early access: November 12, 2024

Published online: November 12, 2024

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Abstract

In this paper, we consider the adaptive boundary stabilization problem for a one-dimensional wave equation with variable coefficients that is subject to general harmonic disturbances at the controlled end. Motivated by the existing related works, this paper develops the adaptive boundary stabilization for the wave equation with variable coefficients in question. First, an adaptive boundary feedback controller is developed in two steps by adaptive and Lyapunov method. Then, it is proved that the closed-loop system is well-posed and asymptotically stable, with the aid of the semigroup approach and LaSalle's invariance principle, respectively. Moreover, the parameter estimates involved in the proposed controller are shown to ultimately converge to their own real values. Finally, the numerical experiments are carried out to show the effectiveness of the proposed scheme.

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Mathematics Subject Classification: Primary: 93C20, 93D15; Secondary: 35L05, 35Q93.

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Figure 1. The solution $ w(x, t) $ of the closed-loop system (65)

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Figure 2. The estimates to the parameters $ \hat{\theta}_{1}(t) $, $ \hat{\varphi}_{1}(t) $ and $ \hat{\theta}_{2}(t) $, $ \hat{\varphi}_{2}(t) $

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