Null Controllability For The Wave Equation With One Moving Endpoint

We investigate in this article the null controllability for the wave equation utt-uxx=0,0<x<k(t),0<t<T with the left endpoint fixed and the right endpoint being moved by k(t),namely, to find a control function acts on the moving endpoint driving the system to rest at time T.By taking a transformation of the variables, we can establish contact with a nonconstant coefficient cylindrical problem to the original non-cylindrical problem and define the solution of the cylindrical problem by applying adjoint equation. Through the establishment of the observability inequality of the adjoint equation, using HUM of J.L.Lions, under some hypotheses of k(t), we can get the final result.