Analysis of exponential stabilization for Rao-Nakra sandwich beam with time-varying weight and time-varying delay: Multiplier method versus observability

Baowei Feng; Carlos Alberto Raposo; Carlos Alberto Nonato; Abdelaziz Soufyane

doi:10.3934/mcrf.2022011

Article Contents

2023, Volume 13, Issue 2: 631-663. Doi: 10.3934/mcrf.2022011

This issue Previous Article Theoretical and computational decay results for a memory type wave equation with variable-exponent nonlinearity Next Article A general maximum principle for partially observed mean-field stochastic system with random jumps in progressive structure

Analysis of exponential stabilization for Rao-Nakra sandwich beam with time-varying weight and time-varying delay: Multiplier method versus observability

1.
Department of Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China
2.
Department of Mathematics, Federal University of São João del-Rei, São João del-Rei, 36307-352, Minas Gerais, Brazil
3.
Department of Mathematics, Federal University of Bahia, Salvador, 40170-115, Bahia, Brazil
4.
Department of Mathematics, College of Sciences, University of Sharjah, P.O.Box 27272, Sharjah, United Arab Emirates

^* Corresponding author: bwfeng@swufe.edu.cn
^* Corresponding author: bwfeng@swufe.edu.cn

Received: July 2021

Revised: February 2022

Early access: March 2022

Published: June 2023

C. A. S. Nonato thanks CAPES (Brazil) for funding the doctoral scholarship

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Abstract

In this paper, we study the global well-posedness and exponential stability for a Rao-Nakra sandwich beam equation with time-varying weight and time-varying delay. The system consists of one Euler-Bernoulli beam equation for the transversal displacement, and two wave equations for the longitudinal displacements of the top and bottom layers. By using the semigroup theory, we show that the system is globally well posed. We give two approaches to obtain the exponential stability. The first one is established by multiplier approach provided the coefficients of delay terms are small. We can also obtain the stability by establishing an equivalence between the stabilization of this system and the observability of the corresponding undamped system. The result is new and is the first result of observability on the Rao-Nakra sandwich beam with with time-varying weight and time-varying delay.

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Mathematics Subject Classification: Primary: 35B40, 93D15; Secondary: 93D20.

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