This paper proposes feedback control strategies for the mitigation of torsional stick-slip oscillations in drilling systems using drag bits. Herein, we employ a model for the coupled axial–torsional drill-string dynamics in combination with a rate-independent bit-rock interaction law including both cutting and frictional effects. Using a singular perturbation and averaging approach, we show that the dynamics of this model generate an apparent velocity-weakening effect in the torque-on-bit, explaining the onset of torsional stick-slip vibrations. Based on this dynamic analysis, the (decoupled) torsional dynamics can be described by a delay-differential equation with a state-dependent delay. Using this model, we propose both state- and output-feedback control strategies for the mitigation of torsional stick-slip oscillations, where the latter strategy uses surface measurements only. The effectiveness of the proposed approaches is shown in a simulation study.