Stabilization with decay estimate of infinite dimensional second order systems by bilinear control damping

Khalil El Kazoui; Hassan Ezzaki; Abed Boulouz

doi:10.3934/eect.2024065

Article Contents

2025, Volume 14, Issue 3: 494-510. Doi: 10.3934/eect.2024065

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Stabilization with decay estimate of infinite dimensional second order systems by bilinear control damping

1.
LAMA Laboratory, Department of Mathematics, Faculty of Sciences, Ibn zohr University, Agadir, Morocco
2.
LRST Laboratory, Department of Mathematics, Faculty of Sciences, Ibn zohr University, Agadir, Morocco

^*Corresponding author: Hassan Ezzaki
^*Corresponding author: Hassan Ezzaki

Received: June 1, 2024

Revised: October 1, 2024

Early access: November 6, 2024

Published online: November 6, 2024

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Abstract

This paper deals with stabilization and energy decay rates of second-order bilinear systems in a Hilbert space. Namely, we examine a class of bounded feedback control to investigate weak, strong, and exponential stabilization. Firstly, we discuss the well-posedness of mild solutions of the considered systems using the cosine family. Secondly, we provide some sufficient conditions to ensure feedback stabilization for the bilinear systems which are presented in terms of observation estimates. Moreover, we derive an explicit decay estimate of the energy in the case of strong stabilization. Finally, we offer examples with numerical simulations to demonstrate the applicability of our theoretical results.

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Mathematics Subject Classification: 93D15, 93D20.

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Figure 1. The dynamic behavior of the displacement and velocity of the closed-loop (29) with the parameter $ \gamma = 0 $

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Figure 2. The dynamic behavior of the displacement and velocity of (29) with $ \gamma = -1$

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Figure 3. The dynamic behavior of the displacement and velocity of the system (29) with the parameter $ \gamma = -2 $

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Figure 4. The evolution of the energy of the system (29)

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