Strong and exponential stabilization of linear boundary control systems

Abdellah Lourini; Mohamed El Azzouzi; Mohamed Laabissi

doi:10.3934/mcrf.2024023

Article Contents

2025, Volume 15, Issue 2: 528-547. Doi: 10.3934/mcrf.2024023

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Strong and exponential stabilization of linear boundary control systems

1.
Department of mathematics, Faculty of Sciences, Chouaib Doukkali University, BP 20, El Jadida, 2400, Morocco
2.
Department of STIN, National School of Applied Sciences, Chouaib Doukkali University, BP 1166, El Jadida, 2400, Morocco

^*Corresponding author: Abdellah Lourini
^*Corresponding author: Abdellah Lourini

Received: October 2023

Revised: April 2024

Early access: June 2024

Published: June 2025

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Abstract

In this work, we focus on the problem of stabilizing boundary control systems. Studying such boundary control systems is widely recognized as a complex issue. To tackle this issue, we employ a strategy that involves transforming the stabilization problem into a corresponding linear internal control system. By utilizing this transformation, we simplify the analysis of boundary stabilization problems and make the most of the extensive body of existing literature. To achieve exponential stability, we utilize spectral decomposition. Additionally, we explore the strong stabilization of a broad class of Riesz-spectral boundary control problems. We apply our findings to a class of heat equations and provide numerical simulations to illustrate these results.

Keywords:

Mathematics Subject Classification: Primary: 93A10, 93D15; Secondary: 47D06, 34K30.

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Figure 1. Evolution of $ \|x(t)\| $, norm of solution of closed-loop system (21)-(22) with $ x_0(z) = \frac{3\pi^2}{4}\cos(\pi z) $

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Figure 2. Evolution of state $ x(z, t) $ of closed-loop system (21)-(22) with $ x_0(z) = \frac{3\pi^2}{4}\cos(\pi z) $

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Figure 3. Evolution of $ \|x(t)\| $ norm of solution of closed-loop system (21)-(25) with $ x_0(z) = \frac{3\pi^2}{4}\cos(\pi z) $

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Figure 4. Evolution of state $ x(z, t) $ for $ x_0(z) = \frac{-\pi^2}{4}(z-z^2)^2+4(z-z^2)-\frac{\pi^2}{4}-2 $

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