Exact controllability for a degenerate and singular wave equation with moving boundary

This paper is concerned with the exact boundary controllability for a degenerate and singular wave equation in a bounded interval with a moving endpoint. By the multiplier method and using an adapted Hardy-poincaré inequality, we prove direct and inverse inequalities for the solutions of the associated adjoint equation. As a consequence, by the Hilbert Uniqueness Method, we deduce the controllability result of the considered system when the control acts on the moving boundary. Furthermore, improved estimates of the speed of the moving endpoint and the controllability time are obtained.