Exact Controllability of Degenerate Wave Equations with Locally Distributed Control in Moving Boundary Domain

Exact internal controllability of a one-dimensional degenerate wave equation in moving boundary domain where the control acts locally is discussed, and two kinds of irregular controls are considered. The equation characterizes the motion of a string with a fixed endpoint and a moving boundary point. A suitable multiplier, which is based on the multiplier method to estimate the energy function, is chosen to demonstrate that the adjoint system is observable. Exact controllability of the original system is established if the adjoint system is observable. Therefore exact controllability of the equation is obtained if the speed of the moving endpoint is lower than a certain constant.