Exact controllability for a one-dimensional wave equation with the fixed endpoint control

Lizhi Cui,Yu Wang,Yang Jiang

doi:10.1186/s13661-015-0476-4

Abstract

This paper is devoted to the study of the exact 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 a moving one. The control is put on the fixed endpoint. When the speed of the moving endpoint is less than the characteristic speed, by the Hilbert uniqueness method (HUM), exact controllability of this equation is established.

Highlights

The main purpose of this paper is to study the exact controllability of ( . )
In Section, we prove that Hilbert uniqueness method (HUM) works very well for ( . )
2 Preliminaries and main results The goal of this paper is to study the exact controllability of ( . ) in the following sense

Summary

Introduction

Consider the following controlled wave equation in the non-cylindrical domain QkT :. There exists a variety of literature on the controllability and stabilization problems of wave equations in a cylindrical domain. Cavalcanti et al [ ] consider existence, uniqueness, and asymptotic behavior of global regular solutions of the mixed problem of the Kirchhoff nonlinear model for the hyperbolic-parabolic equation in non-cylindrical domains. In [ – ], some controllability results for wave equations with Dirichlet boundary conditions in suitable non-cylindrical domains were investigated, respectively. Some additional conditions on the moving boundary were required, which render the method used in [ ] and [ ] inapplicable to the controllability problems of The key point is to define directly the energy function of a wave equation in the non-cylindrical domain and use the multiplier method to overcome these difficulties.

For k

Findings

From this we get t