  Two-dimensional (2D) flows are efficiently controlled with spanwise waviness,
i.e. spanwise-periodic (SP) wall blowing/suction/deformation. We tackle the
global linear stability of 2D flows subject to small-amplitude 3D SP control.
Building on previous work for parallel flows (Boujo et al. 2015), an adjoint
method is proposed for computing the 2nd-order eigenvalue sensitivity. Such
control has a zero 1st-order linear effect, so the 2nd-order quadratic effect
prevails. The sensitivity operator allows one (i) to predict the effect of any
control without computing the controlled flow, (ii) to compute the optimal
control for growth rate/frequency modification. The method takes advantage of
spanwise periodicity to reduce computational complexity (from a 3D problem to a
2D one). The approach is applied to the leading eigenvalue of the laminar flow
around a circular cylinder. Two SP controls are explored: wall actuation
(blowing/suction) and wall deformation. Decomposing the eigenvalue variation,
we find that the 3D contribution (from the SP 1st-order flow modification) is
generally larger than the 2D one from the mean flow correction
(spanwise-invariant 2nd-order flow modification). The optimal control for
stabilization is top-down symmetric, leading to varicose streaks in the wake.
Analyzing the competition between amplification and stabilization shows that
optimal varicose streaks are not significantly more amplified than sinuous ones
but have a stronger stabilizing effect. The optimal wall deformation induces a
flow modification similar to that of the optimal wall actuation. In general,
spanwise and tangential actuation have a small contribution to the optimal
control, so normal-only actuation is a good trade-off between simplicity and
effectiveness. Our method opens the way to the systematic design of optimal SP
control for a variety of control objectives other than linear stability.
