Exact controllability for a wave equation with fixed boundary control

Lizhi Cui,Libo Song

doi:10.1186/1687-2770-2014-47

Lizhi Cui, Libo Song

Open Access

https://doi.org/10.1186/1687-2770-2014-47

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Abstract

This paper addresses the study of the controllability for a one-dimensional wave equation in domains with moving boundary. This equation characterizes the motion of a string with a fixed endpoint and the other one moving. When the speed of the moving endpoint is less than , by the Hilbert Uniqueness Method, the exact controllability of this equation is established. Also, the explicit dependence of the controllability time on the speed of the moving endpoint is given.

Highlights

There are only a few works on the exact controllability for wave equations defined in non-cylindrical domains
In [ ], the exact controllability of a multi-dimensional wave equation with constant coefficients in a non-cylindrical domain was established, while the control entered the system through the whole non-cylindrical domain
In Section, we prove that the Hilbert Uniqueness Method (HUM) works very well for ( . )

Summary

Introduction

Consider the following controlled wave equation in the non-cylindrical domain QkT :. ). As we all know, there exists much literature on the controllability problems of wave equations in a cylindrical domain. There are only a few works on the exact controllability for wave equations defined in non-cylindrical domains. In [ ] and [ ], some controllability results for the wave equations with Dirichlet boundary conditions in suitable non-cylindrical domains were investigated, respectively. Some additional conditions on the moving boundary were required, which entail the method used in [ ] and [ ] not to be applicable to the controllability problems of We mainly use the multiplier method to overcome these difficulties and drop the additional conditions for the moving boundary. The main result of this paper is stated as follows

For any given

Suppose that

Findings

It follows that