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Journal of Applied Nonlinear Dynamics 11(2) (2022) 343--358 | DOI:10.5890/JAND.2022.06.006

Abdelkarim Kelleche, Nasser-eddine Tatar

Facult'{e} des Sciences et de la Technologie, Universit'{e} Djilali Boun^{a}ama, Algeria

normalsize Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran

31261, Saudi Arabia

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Abstract

This paper addresses the stabilization question for a nonlinear model of an axially moving string. The string is tensioned and is subject to spatiotemporary varying disturbances. The Hamilton principle of changing mass is employed to formulate mathematically the problem. By means of the Faedo--Galerkin method, we establish the well-posedness. A boundary control with a time-varying delay is designed to stabilize uniformly the string. Then, we derive a decay rate of the solution under the condition that the retarded term be dominated by the damping one. Some examples are given to clarify when the rate is exponential or polynomial.

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