We provide in-depth scrutiny of two methods making use of adjoint-based gradients to compute the sensitivity of drag in the two-dimensional, periodic flow past a circular cylinder (Re≲189): first, the time-stepping analysis used in Meliga et al. [Phys. Fluids 26, 104101 (2014)] that relies on classical Navier-Stokes modeling and determines the sensitivity to any generic control force from time-dependent adjoint equations marched backwards in time; and, second, a self-consistent approach building on the model of Mantič-Lugo et al. [Phys. Rev. Lett. 113, 084501 (2014)] to compute semilinear approximations of the sensitivity to the mean and fluctuating components of the force. Both approaches are applied to open-loop control by a small secondary cylinder and allow identifying the sensitive regions without knowledge of the controlled states. The theoretical predictions obtained by time-stepping analysis reproduce well the results obtained by direct numerical simulation of the two-cylinder system. So do the predictions obtained by self-consistent analysis, which corroborates the relevance of the approach as a guideline for efficient and systematic control design in the attempt to reduce drag, even though the Reynolds number is not close to the instability threshold and the oscillation amplitude is not small. This is because, unlike simpler approaches relying on linear stability analysis to predict the main features of the flow unsteadiness, the semilinear framework encompasses rigorously the effect of the control on the mean flow, as well as on the finite-amplitude fluctuation that feeds back nonlinearly onto the mean flow via the formation of Reynolds stresses. Such results are especially promising as the self-consistent approach determines the sensitivity from time-independent equations that can be solved iteratively, which makes it generally less computationally demanding. We ultimately discuss the extent to which relevant information can be gained from a hybrid modeling computing self-consistent sensitivities from the postprocessing of DNS data. Application to alternative control objectives such as increasing the lift and alleviating the fluctuating drag and lift is also discussed.