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December  2018, 7(4): 599-616. doi: 10.3934/eect.2018029

## Existence and stabilization of a Kirchhoff moving string with a delay in the boundary or in the internal feedback

 1 Faculé des sciences et de la technologie université Djilali Bounaama, Route Theniet El Had, Soufay 44225 Khemis Miliana, Algeria 2 Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

* Corresponding author: Abdelkarim Kelleche

Received  June 2017 Revised  August 2018 Published  September 2018

Fund Project: The second author is supported by King Fahd University of Petroleum and Minerals through the Project no.: IN151015.

In this paper, we study the effect of an internal or boundary time-delay on the stabilization of a moving string. The models adopted here are nonlinear and of "Kirchhoff" type. The well-posedness of the systems is proven by means of the Faedo-Galerkin method. In both cases, we prove that the solution of the system approaches the equilibrium in an exponential manner in the energy norm. To this end we request that the delayed term be dominated by the damping term. This is established through the multiplier technique.

Citation: Abdelkarim Kelleche, Nasser-Eddine Tatar. Existence and stabilization of a Kirchhoff moving string with a delay in the boundary or in the internal feedback. Evolution Equations & Control Theory, 2018, 7 (4) : 599-616. doi: 10.3934/eect.2018029
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