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For a Mach 
$4.5$ flat-plate adiabatic boundary layer, we study the sensitivity of the first, second Mack modes and streaks to steady wall-normal blowing/suction and wall heat flux. The global instabilities are characterised in frequency space with resolvent gains and their gradients with respect to wall-boundary conditions are derived through a Lagrangian-based method. The implementation is performed in the open-source high-order finite-volume code BROADCAST and algorithmic differentiation is used to access the high-order state derivatives of the discretised governing equations. For the second Mack mode, the resolvent optimal gain decreases when suction is applied upstream of Fedorov's mode 
$S$/mode 
$F$ synchronisation point, leading to stabilisation, and the converse when applied downstream. The largest suction gradient is in the region of branch I of mode 
$S$ neutral curve. For heat-flux control, strong heating at the leading edge stabilises both the first and second Mack modes, the former being more sensitive to wall-temperature control. Streaks are less sensitive to any boundary control in comparison with the Mack modes. Eventually, we show that an optimal actuator consisting of a single steady heating strip located close to the leading edge manages to damp the linear growth of all three instability mechanisms.
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 us 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 us 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 set-ups, i.e. with estimation sensor placed respectively upstream/downstream of the actuator, allow us 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 set-up 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 set-up for performance robustness in convectively unstable flows.
