The paper considers oilwell drilling operations and, in particular, the problem of minimizing drilling vibrations. We develop a stabilizing controller for a one dimension wave PDE with damping term that describes the drillstring torsional system. The drillstring system stability is studied via the Lyapunov theory and the control steps are deduced from a PDE backstepping approach. As a result, exponential vibration suppression is obtained in presence of stick-slip friction due to the drill-bit and the soil interactions. The torsional vibration rejection is shown through the developed simulation results.