We assess experimentally and theoretically the ability of a small control cylinder to alter vortex shedding in turbulent flow past a square cylinder at R e = 22,000. Results are presented in terms of sensitivity maps showing the flow regions where the shedding frequency and amplitude are most affected by the control cylinder. Experimental results are obtained for a ratio 0.02 of the cylinder diameters, over an extended domain covering the wake, the shear layers and the free stream. The shedding frequency can be either decreased or increased, the largest effects being obtained placing the control cylinder at the outer edge of the detached shear layers (associated with frequency decrease) or upstream of the square cylinder (associated with frequency increase, in contrast with previous results obtained for a D-shaped geometry of the main cylinder). In contrast, the oscillation amplitude is rarely decreased, meaning that any variation of the shedding frequency comes at the expense of more intense vortex shedding. These findings are revisited in the frame of a theoretical, linear sensitivity analysis of the time-averaged mean flow, performed using adjoint methods in the frame of Reynolds-averaged Navier-Stokes modeling. We show that the retained approach carries valuable information in view of guiding efficient control strategy, as it allows identifying the main regions yielding either a decrease or an increase of the shedding frequency in striking agreement with the experiments. This is a tremendous timesaving in so far as the controlled states need not be computed, the overall computational cost being roughly that of computing the mean flow. In contrast, performing the sensitivity analysis on the underlying unstable steady state yields flawed predictions, hence stressing the need to encompass some level of mean coherent-coherent perturbations interaction in the linear model.