We consider the control of nominally two-dimensional (2D) shear flows invariant in the spanwise direction, by three-dimensional (3D) spanwise-periodic perturbations associated with streamwise streaks. For this type of control the first-order sensitivity of the eigenvalues to the control amplitude is zero and the leading order variation of the eigenvalues depends quadratically on the streaks' amplitude A s as µ 3D − µ 2D ∼ A 2 s µ 2. In many previous studies aimed at understanding the stabilizing mechanism of 3D perturbations and/or explicitly computing the second-order sensitivity µ 2 [3, 2, 4, 1] only the effect of the first spanwise harmonic (the 'linear streaks') of the basic flow distortion was included in the analysis. As the linear streaks are of amplitude A s , these analyses considered only a first-order perturbation of the linear stability operator L 3D ∼ L 2D + A s L. When the streaks are forced starting with streamwise vortices, however, the Reynolds stresses induce a spanwise uniform (0th spanwise harmonic) deformation of the basic flow. The inclusion of this O(A 2 s) deformation in the analysis (i.e. L 3D ∼ L 2D + A s L + A 2 s L) leads to a 'composite' and consistent second-order sensitivity µ 2 = µ 2 + µ 2 (where the additional term µ 2 was neglected in previous analyses). In this study we address this issue using as a testbed for the analysis the 3D control of the Kelvin-Helmholtz instability developing in the (2D) parallel mixing layer with hyperbolic-tangent profile. We compare the results obtained using as basic flow the 3D profile U 3D (y, z) issued from the nonlinear Navier-Stokes equations without any assumption to (a) those, associated to µ 2 , obtained using a first-harmonic approximation on the basic flow U SA = U 2D (y) + A s u 1 (y) cos βz, (b) those, associated to µ 2 , obtained including only the spanwise uniform part of the basic flow distortion U (y) = U 2D (y) + ∆U (y) and (c) those obtained including only the spanwise varying part of the basic flow distortion U (y, z) = U 2D (y) + ∆U (y, z) The main findings, illustrated in Figure 1, can be summarized as follows: (1) the sensitivities based on U SA (magenta) and on U (red) are very similar, (2) the sensitivities based on U SA and U have almost always a sign opposite to those based on U 3D (black), while the opposite is true for those (blue) based on U , (3) the composite second order sensitivities µ 2 = µ 2 + µ 2 based on the inclusion of the first and zero-th harmonics in the linear operator (green) are in good agreement with those based on U 3D (black) for small to moderate values of A s .