Fractional derivative boundary control in coupled Euler-Bernoulli beams: stability and discrete energy decay

Boumediene Boukhari; Foued Mtiri; Ahmed Bchatnia; Abderrahmane Beniani; Boumediene Boukhari; Foued Mtiri; Ahmed Bchatnia; Abderrahmane Beniani

doi:10.3934/math.20241541

AIMS Mathematics

2024, Volume 9, Issue 11: 32102-32123. doi: 10.3934/math.20241541

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Fractional derivative boundary control in coupled Euler-Bernoulli beams: stability and discrete energy decay

1.
Department of Mathematics, Faculty of Science and Technology, Engineering and Sustainable Development Laboratory, University of Ain Temouchent, Ain Témouchent, 46000, Algeria
2.
Department of Mathematics, Applied College in Mahayil, King Khalid University, Abha, Saudi Arabia
3.
Department of Mathematics, Faculty of Sciences of Tunis, LR Analyse Non-Linéaire et Géométrie, LR21ES08, University of Tunis El Manar, Tunis, 2092, Tunisia

Received: 06 September 2024 Revised: 30 October 2024 Accepted: 30 October 2024 Published: 12 November 2024
MSC : 26A33, 35L55, 74D05, 93D15, 93D20

This paper analyzes an Euler-Bernoulli beam equation in a bounded domain with a boundary control condition involving a fractional derivative. By utilizing the semigroup theory of linear operators and building on the results of Borichev and Tomilov, the stability properties of the system are examined. Additionally, a numerical scheme is developed to reproduce various decay rate behaviors. The numerical simulations confirm the theoretical stability results regarding the energy decay rate and demonstrate exponential decay for specific configurations of initial data.
- Euler-Bernoulli beam equation,
- dynamic boundary dissipation of fractional derivative type,
- frequency domain method,
- strong stability
Citation: Boumediene Boukhari, Foued Mtiri, Ahmed Bchatnia, Abderrahmane Beniani. Fractional derivative boundary control in coupled Euler-Bernoulli beams: stability and discrete energy decay[J]. AIMS Mathematics, 2024, 9(11): 32102-32123. doi: 10.3934/math.20241541

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Abstract

This paper analyzes an Euler-Bernoulli beam equation in a bounded domain with a boundary control condition involving a fractional derivative. By utilizing the semigroup theory of linear operators and building on the results of Borichev and Tomilov, the stability properties of the system are examined. Additionally, a numerical scheme is developed to reproduce various decay rate behaviors. The numerical simulations confirm the theoretical stability results regarding the energy decay rate and demonstrate exponential decay for specific configurations of initial data.

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