  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
conversely 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 all the
instabilities together.
