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

Abdelkarim Kelleche; Nasser-Eddine Tatar

doi:10.3934/eect.2018029

Article Contents

2018, Volume 7, Issue 4: 599-616. Doi: 10.3934/eect.2018029

This issue Previous Article Some partially observed multi-agent linear exponential quadratic stochastic differential games Next Article Backward controllability of pullback trajectory attractors with applications to multi-valued Jeffreys-Oldroyd equations

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

Abdelkarim Kelleche^1, , and
Nasser-Eddine Tatar^2,

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
* Corresponding author: Abdelkarim Kelleche

Received: June 2017

Revised: August 2018

Published: September 2018

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 35L20, 93D15; Secondary: 93D20.

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