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June  2017, 10(3): 557-579. doi: 10.3934/dcdss.2017028

## Exponential stability of 1-d wave equation with the boundary time delay based on the interior control

 Department of Mathematics, Tianjin University, Haihe Education Park, Tianjin, Tianjin 300350, China

Received  June 2016 Revised  January 2017 Published  February 2017

Fund Project: This work is supported by Science Foundation of China under Grant Nos.61174080, 61503275 and 61573252

In this paper, the stability problem of 1-d wave equation with the boundary delay and the interior control is considered. The well-posedness of the closed-loop system is investigated by the linear operator. Based on the idea of Lyapunov functional technology, we give the condition on the relationship between the control parameter α and the delay parameter k to guarantee the exponential stability of the system.

Citation: Yaru Xie, Genqi Xu. Exponential stability of 1-d wave equation with the boundary time delay based on the interior control. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 557-579. doi: 10.3934/dcdss.2017028
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