  The question of optimal spanwise-periodic modification for the stabilisation
of spanwise-invariant flows is addressed. A 2nd-order sensitivity analysis is
conducted for the linear temporal stability of parallel flows U0 subject to
small-amplitude spanwise-periodic modification e*U1, e<<1. Spanwise-periodic
modifications have a quadratic effect on stability, i.e. the 1st-order
eigenvalue variation is zero. A 2nd-order sensitivity operator is computed from
a 1D calculation, allowing one to predict how eigenvalues are affected by any
U1, without actually solving for modified eigenvalues/eigenmodes. Comparisons
with full 2D stability calculations in a plane channel flow and in a mixing
layer show excellent agreement. Next, optimisation is performed on the
2nd-order sensitivity operator: for each eigenmode streamwise wavenumber and
base flow modification spanwise wavenumber b, the most stabilising profiles U1
are computed, together with lower bounds for the variation in leading
eigenvalue. These bounds increase like b^-2 as b goes to 0, yielding a large
stabilising potential. However, 3D modes with wavenumbers |b0|=b and b/2 are
destabilised, thus larger control wavenumbers should be preferred. The
modification U1 optimised for the most unstable streamwise wavenumber has a
stabilising effect on other streamwise wavenumbers too. Finally, the potential
of transient growth to amplify perturbations and stabilise the flow is
assessed. Combined optimal perturbations that achieve the best balance between
transient linear amplification and flow stabilisation are determined. In the
mixing layer with b<1.5, these combined optimal perturbations appear similar to
transient growth-only optimal perturbations, and achieve a more efficient
overall stabilisation than optimal 1D and 2D modifications computed for
stabilisation only. This is consistent with the efficiency of streak-based
control strategies.
