ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Existence and Boundary Control of a Nonlinear Axially Moving String Subject to Disturbances

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\addressNewline 31261, Saudi Arabia

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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