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doi: 10.3934/dcdss.2021090
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## 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 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 & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021090
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