  Streamwise vortices and the associated streaks evolve in boundary layers over
flat or concave surfaces due to disturbances initiated upstream or triggered by
the wall surface. Following the transient growth phase, the fully-developed
vortex structures become susceptible to inviscid secondary instabilities
resulting in early transition to turbulence via `bursting' processes. In
high-speed boundary layers, more complications arise due to compressibility and
thermal effects, which become more significant for higher Mach numbers. In this
paper, we study G\"{o}rtler vortices developing in high-speed boundary layers
using the boundary region equations (BRE) formalism, which we solve using an
efficient numerical algorithm. Streaks are excited using a small transpiration
velocity at the wall. Our BRE-based algorithm is found to be superior to direct
numerical simulation (DNS) and ad-hoc nonlinear parabolized stability equation
(PSE) models. BRE solutions are less computationally costly than a full DNS and
have a more rigorous theoretical foundation than PSE-based models. For example,
the full development of a G\"{o}rtler vortex system in high-speed boundary
layers can be predicted in a matter of minutes using a single processor via the
BRE approach. This substantial reduction in calculation time is one of the
major achievements of this work. We show, among other things, that it allows
investigation into feedback control in reasonable total computational times. We
investigate the development of the G\"{o}rtler vortex system via the BRE
solution with feedback control parametrically at various freestream Mach
numbers $M_\infty$ and spanwise separations $\lambda$ of the inflow
disturbances.
