  The two-dimensional backward-facing step flow is a canonical example of noise
amplifier flow: global linear stability analysis predicts that it is stable,
but perturbations can undergo large amplification in space and time as a result
of non-normal effects. This amplification potential is best captured by optimal
transient growth analysis, optimal harmonic forcing, or the response to
sustained noise. In view of reducing disturbance amplification in these
globally stable open flows, a variational technique is proposed to evaluate the
sensitivity of stochastic amplification to steady control. Existing sensitivity
methods are extended in two ways to achieve a realistic representation of
incoming noise: (i) perturbations are time-stochastic rather than
time-harmonic, (ii) perturbations are localised at the inlet rather than
distributed in space. This allows for the identification of regions where
small-amplitude control is the most effective, without actually computing any
controlled flows. In particular, passive control by means of a small cylinder
and active control by means of wall blowing/suction are analysed for Reynolds
number $Re=500$ and step-to-outlet expansion ratio $\Gamma=0.5$. Sensitivity
maps for noise amplification appear largely similar to sensitivity maps for
optimal harmonic amplification at the most amplified frequency. This is
observed at other values of $Re$ and $\Gamma$ too, and suggests that the design
of steady control in this noise amplifier flow can be simplified by focusing on
the most dangerous perturbation at the most dangerous frequency.
