This article is largely concerned with the design of trajectory tracking controllers for a class of frictional oscillators, represented by differential inclusions with maximal monotone operators. A backstepping passivity-based control strategy is proposed, which relies on set-valued friction and its monotonicity properties (hence the goal is not to compensate for friction but to take advantage of it). Both the continuoustime design and its discrete-time counterpart are analysed. Robustness with respect to uncertainties in the friction coefficient is carefully analyzed in both continuous and discrete time settings. Details about the set-valued controller implementation are provided. Numerical simulations illustrate the theoretical developments.