Uniform stabilization of a wave equation with partial Dirichlet delayed control

Xiaorui Wang; Genqi Xu

doi:10.3934/eect.2020022

Article Contents

2020, Volume 9, Issue 2: 509-533. Doi: 10.3934/eect.2020022

This issue Previous Article On a Kirchhoff wave model with nonlocal nonlinear damping Next Article Null controllability for a heat equation with dynamic boundary conditions and drift terms

Uniform stabilization of a wave equation with partial Dirichlet delayed control

Xiaorui Wang^1,2, , and
Genqi Xu^1,

1.
School of Mathematics, Tianjin University, Tianjin 300354, China
2.
Department of Mathematics and Statistics, Qinghai Nationalities University, Xining, Qinghai 810007, China

^* Corresponding author: Xiaorui Wang
^* Corresponding author: Xiaorui Wang

Received: October 31, 2018

Revised: June 30, 2019

Early access: December 2019

Published: June 2020

This project is partially supported by the National Natural Science Foundation in China (NSFC 61773277), and partially supported by NSF of Qinghai Province (2017-ZJ-908)

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Abstract

In this paper, we consider the uniform stabilization of some high-dimensional wave equations with partial Dirichlet delayed control. Herein we design a parameterization feedback controller to stabilize the system. This is a new approach of controller design which overcomes the difficulty in stability analysis of the closed-loop system. The detailed procedure is as follows: At first we rewrite the system with partial Dirichlet delayed control into an equivalence cascaded system of a transport equation and a wave equation, and then we construct an exponentially stable target system; Further, we give the form of the parameterization feedback controller. To stabilize the system under consideration, we choose some appropriate kernel functions and define a bounded inverse linear transformation such that the closed-loop system is equivalent to the target system. Finally, we obtain the stability of closed-loop system by the stability of target system.

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Mathematics Subject Classification: Primary: 93C20, 93D15; Secondary: 35G15, 35L05.

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