  Adjoint-based sensitivity analysis is routinely used today to assess
efficiently the effect of open-loop control on the linear stability properties
of unstable flows. Sensitivity maps identify regions where small-amplitude
control is the most effective, i.e. yields the largest first-order (linear)
eigenvalue variation. In this study an adjoint method is proposed for computing
a second-order (quadratic) sensitivity operator, and applied to the flow past a
circular cylinder, controlled with a steady body force or a passive device
model. Maps of second-order eigenvalue variations are obtained, without
computing controlled base flows and eigenmodes. For finite control amplitudes,
the second-order analysis improves the accuracy of the first-order prediction,
and informs about its range of validity, and whether it underestimates or
overestimates the actual eigenvalue variation. Regions are identified where
control has little or no first-order effect but a second-order effect. In the
cylinder wake, the effect of a control cylinder tends to be underestimated by
the first-order sensitivity, and including second-order effects yields larger
regions of flow restabilisation. Second-order effects can be decomposed into
two mechanisms: second-order base flow modification, and interaction between
first-order modifications of the base flow and eigenmode. Both contribute
equally in general in sensitive regions of the cylinder wake. Exploiting the
second-order sensitivity operator, the optimal control maximising the total
second-order stabilisation is computed via a quadratic eigenvalue problem. The
approach is applicable to other types of control (e.g. wall blowing/suction and
shape deformation) and other eigenvalue problems (e.g. amplification of
time-harmonic perturbations, or resolvent gain, in stable flows).
