  G\"{o}rtler vortices in boundary layer flows over concave surfaces are caused
by the imbalance between centrifugal effects and radial pressure gradients.
Depending on various geometrical and/or freestream flow conditions, vortex
breakdown via secondary instabilities leads to early transition to turbulence.
It is desirable, therefore, to reduce vortex energy in an attempt to delay the
transition from laminar to turbulent flow, and thereby achieve a reduced
frictional drag. To this end, we apply a proportional control algorithm aimed
at reducing the wall shear stress and the energy of G\"{o}rtler vortices
evolving in high-speed boundary layers. The active control scheme is based on
wall transpiration with sensors placed either in the flow or at the wall. In
addition, we evaluate the effect of wall cooling and heating on G\"{o}rtler
vortices evolving in high-speed boundary layers, by reducing or increasing the
upstream wall temperature. The numerical results are obtained by solving the
full Navier-Stokes equations in generalized curvilinear coordinates, using a
high-order numerical algorithm. Our results show that the active control based
on wall transpiration reduces both the wall shear stress and the energy of the
G\"{o}rtler vortices; the passive control based on wall cooling or heating
reduces the wall shear stress, but slightly increases the energy of the
vortices in both supersonic and hypersonic regimes.
