In drilling operations, the drillstring interaction with the borehole gives rise to a wide variety of non-desired oscillations. The main types of drilling vibrations are torsional (stick-slip), axial (bit-bounce) and lateral (whirling). The analysis and modeling of rotary drilling vibrations is a topic whose economical interest has been renewed by recent oil fields discoveries leading to a growing literature. This contribution addresses the problem of modeling and control of the most critical vibration mode: the stick-slip phenomenon. The wave equation subject to mixed boundary conditions is used to reproduce torsional oscillations of the drillstring. By means of a direct transformation we derive an input-output model described by a neutral-type time-delay equation which clearly simplifies the system simulation. A dissipativity analysis of the wave equation model leads to the design of a stabilizing controller guaranteeing a non-growth of energy during the drilling process. Furthermore, by means of a polytopic representation of the neutral-type time-delay model, stability and stabilization conditions in terms of Linear Matrix Inequalities (LMIs) are given. The performance of the proposed approaches is highlighted through simulations showing an effective suppression of the stick-slip phenomenon.