EXACT CONTROLLABILITY PROBLEM OF A WAVE EQUATION IN NON-CYLINDRICAL DOMAINS

Let alpha : [0, infinity) > (0, infinity) be a twice continuous differentiable function which satisfies that alpha(0) = 1, alpha' is monotone and 0 < c(1) <= alpha'(t) (t) <= c(2) < 1 for some constants c(1), c(2). The exact controllability of a one-dimensional wave equation in a non-cylindrical domain is proved. This equation characterizes small vibrations of a string with one of its endpoint fixed and the other moving with speed alpha'(t). By using the Hilbert Uniqueness Method, we obtain the exact controllability results of this equation with Dirichlet boundary control on one endpoint. We also give an estimate on the controllability time that depends only on c(1) and c(2).