# American Institute of Mathematical Sciences

May  2022, 15(5): 1127-1141. doi: 10.3934/dcdss.2021090

## Boundary stabilization of a flexible structure with dynamic boundary conditions via one time-dependent delayed boundary control

 1 Kuwait University, Faculty of Science, Department of Mathematics, Safat 13060, Kuwait 2 UR Analysis and Control of PDEs, UR13ES64, Department of Mathematics, Faculty of Sciences of Monastir, University of Monastir, 5019 Monastir, Tunisia

* Corresponding author

Received  March 2021 Revised  June 2021 Published  May 2022 Early access  August 2021

This article deals with the dynamic stability of a flexible cable attached at its top end to a cart and a load mass at its bottom end. The model is governed by a system of one partial differential equation coupled with two ordinary differential equations. Assuming that a time-dependent delay occurs in one boundary, the main concern of this paper is to stabilize the dynamics of the cable as well as the dynamical terms related to the cart and the load mass. To do so, we first prove that the problem is well-posed in the sense of semigroups theory provided that some conditions on the delay are satisfied. Thereafter, an appropriate Lyapunov function is put forward, which leads to the exponential decay of the energy as well as an estimate of the decay rate.

Citation: Boumedièene Chentouf, Sabeur Mansouri. Boundary stabilization of a flexible structure with dynamic boundary conditions via one time-dependent delayed boundary control. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1127-1141. doi: 10.3934/dcdss.2021090
##### References:

show all references

##### References:
The platform
 [1] Yaru Xie, Genqi Xu. Exponential stability of 1-d wave equation with the boundary time delay based on the interior control. Discrete and Continuous Dynamical Systems - S, 2017, 10 (3) : 557-579. doi: 10.3934/dcdss.2017028 [2] Serge Nicaise, Cristina Pignotti, Julie Valein. Exponential stability of the wave equation with boundary time-varying delay. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 693-722. doi: 10.3934/dcdss.2011.4.693 [3] Aowen Kong, Carlos Nonato, Wenjun Liu, Manoel Jeremias dos Santos, Carlos Raposo. Equivalence between exponential stabilization and observability inequality for magnetic effected piezoelectric beams with time-varying delay and time-dependent weights. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 2959-2978. doi: 10.3934/dcdsb.2021168 [4] Masahiro Kubo, Noriaki Yamazaki. Periodic stability of elliptic-parabolic variational inequalities with time-dependent boundary double obstacles. Conference Publications, 2007, 2007 (Special) : 614-623. doi: 10.3934/proc.2007.2007.614 [5] Mourad Bellassoued, Oumaima Ben Fraj. Stability estimates for time-dependent coefficients appearing in the magnetic Schrödinger equation from arbitrary boundary measurements. Inverse Problems and Imaging, 2020, 14 (5) : 841-865. doi: 10.3934/ipi.2020039 [6] István Györi, Ferenc Hartung. Exponential stability of a state-dependent delay system. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 773-791. doi: 10.3934/dcds.2007.18.773 [7] Yuyun Zhao, Yi Zhang, Tao Xu, Ling Bai, Qian Zhang. pth moment exponential stability of hybrid stochastic functional differential equations by feedback control based on discrete-time state observations. Discrete and Continuous Dynamical Systems - B, 2017, 22 (1) : 209-226. doi: 10.3934/dcdsb.2017011 [8] Mourad Choulli, Yavar Kian. Stability of the determination of a time-dependent coefficient in parabolic equations. Mathematical Control and Related Fields, 2013, 3 (2) : 143-160. doi: 10.3934/mcrf.2013.3.143 [9] Guanghui Hu, Yavar Kian. Uniqueness and stability for the recovery of a time-dependent source in elastodynamics. Inverse Problems and Imaging, 2020, 14 (3) : 463-487. doi: 10.3934/ipi.2020022 [10] Eduard Marušić-Paloka, Igor Pažanin. Reaction of the fluid flow on time-dependent boundary perturbation. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1227-1246. doi: 10.3934/cpaa.2019059 [11] Ismael Maroto, Carmen Núñez, Rafael Obaya. Exponential stability for nonautonomous functional differential equations with state-dependent delay. Discrete and Continuous Dynamical Systems - B, 2017, 22 (8) : 3167-3197. doi: 10.3934/dcdsb.2017169 [12] Yi Zhang, Yuyun Zhao, Tao Xu, Xin Liu. $p$th Moment absolute exponential stability of stochastic control system with Markovian switching. Journal of Industrial and Management Optimization, 2016, 12 (2) : 471-486. doi: 10.3934/jimo.2016.12.471 [13] Xiang Xie, Honglei Xu, Xinming Cheng, Yilun Yu. Improved results on exponential stability of discrete-time switched delay systems. Discrete and Continuous Dynamical Systems - B, 2017, 22 (1) : 199-208. doi: 10.3934/dcdsb.2017010 [14] Soumia Saïdi, Fatima Fennour. Second-order problems involving time-dependent subdifferential operators and application to control. Mathematical Control and Related Fields, 2022  doi: 10.3934/mcrf.2022019 [15] Xiongxiong Bao, Wan-Tong Li. Existence and stability of generalized transition waves for time-dependent reaction-diffusion systems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3621-3641. doi: 10.3934/dcdsb.2020249 [16] Jiangshan Wang, Lingxiong Meng, Hongen Jia. Numerical analysis of modular grad-div stability methods for the time-dependent Navier-Stokes/Darcy model. Electronic Research Archive, 2020, 28 (3) : 1191-1205. doi: 10.3934/era.2020065 [17] Alexander Zlotnik, Ilya Zlotnik. Finite element method with discrete transparent boundary conditions for the time-dependent 1D Schrödinger equation. Kinetic and Related Models, 2012, 5 (3) : 639-667. doi: 10.3934/krm.2012.5.639 [18] Zhiqing Liu, Zhong Bo Fang. Blow-up phenomena for a nonlocal quasilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3619-3635. doi: 10.3934/dcdsb.2016113 [19] Martin Kružík, Johannes Zimmer. Rate-independent processes with linear growth energies and time-dependent boundary conditions. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 591-604. doi: 10.3934/dcdss.2012.5.591 [20] Fuke Wu, George Yin, Le Yi Wang. Razumikhin-type theorems on moment exponential stability of functional differential equations involving two-time-scale Markovian switching. Mathematical Control and Related Fields, 2015, 5 (3) : 697-719. doi: 10.3934/mcrf.2015.5.697

2021 Impact Factor: 1.865