  We consider closed-loop control of a two-dimensional supersonic boundary
layer at M = 4.5 that aims at reducing the linear growth of second Mack mode
instabilities. These instabilities are first characterized with local spatial
and global resolvent analyses, which allow to refine the control strategy and
to select appropriate actuators and sensors. After linear input-output reduced
order models have been identified, multi-criteria structured mixed
$H_{2}/H_{\infty}$ synthesis allows to fix beforehand the controller structure
and to minimize appropriate norms of various transfer functions: the $H_{2}$
norm to guarantee performance (reduction of perturbation amplification in
nominal condition) and the $H_{\infty}$ norm to maintain performance robustness
(with respect to sensor noise) and stability robustness (with respect to
uncertain free-stream velocity/density variations). Both feedforward and
feedback setups, i.e. with estimation sensor placed respectively
upstream/downstream of the actuator, allow to maintain the local perturbation
energy below a given threshold over a significant distance downstream of the
actuator, even in the case of noisy estimation sensors or free-stream density
variations. However, the feedforward setup becomes completely ineffective when
convective time-delays are altered by free-stream velocity variations of $\pm
5\%$, which highlights the strong relevance of the feedback setup for
performance robustness in convectively unstable flows.
