In oilwell drillstring systems, vibrations represent an important source of economic losses; drill bit wear, pipes disconnection, borehole disruption and prolonged drilling time, are only a few examples of consequences associated with drilling vibrations. In this paper a control-oriented model of torsional vibrations occurring in rotary drilling process is proposed. More precisely, a wave equations with weak damping term is considered. An appropriate stabilizing controller with a reduced number of parameters is proposed for damping such torsional vibrations. Such a controller allows to further explore the effect of multiple roots with maximal admissible multiplicity for linear neutral system with a single delay. An illustrative example completes the presentation. © 2019 American Automatic Control Council.