Existence and controllability for a one-dimensional wave equation in a non-cylindrical domain

We consider a finite string vibrating described by a one-dimensional wave equation. The left boundary point of the string is fixed, while the right boundary point is moving. The controls are put on both end points. Assume that speed of the moving end be equal with the characteristic speed of the wave equation. We shall prove existence of weak solution for the wave equation of Cauchy-Goursat type. Moreover, sufficient conditions which ensure the exact controllability are formulated by the specific expression of the weak solution.