  The control of stream-wise vortices in high Reynolds number boundary layer
flows often aims at reducing the vortex energy as a means of mitigating the
growth of secondary instabilities, which eventually delay the transition from
laminar to turbulent flow. In this paper, we aim at utilizing such an energy
reduction strategy using optimal control theory to limit the growth of
G\"{o}rtler vortices developing in an incompressible laminar boundary layer
flow over a concave wall, and excited by a row of roughness elements with
span-wise separation in the same order of magnitude as the boundary layer
thickness. Commensurate with control theory formalism, we transform a
constrained optimization problem into an unconstrained one by applying the
method of Lagrange multipliers. A high Reynolds number asymptotic framework is
utilized, wherein the Navier-Stokes equations are reduced to the boundary
region equations (BRE), in which wall deformations enter the problem through an
appropriate Prandtl transformation. In the optimal control strategy, the wall
displacement or the wall transpiration velocity serve as control variables,
while the cost functional is defined in terms of the wall shear stress. Our
numerical results indicate, among other things, that the optimal control
algorithm is very effective in reducing the amplitude of the G\"{o}rtler
vortices, especially for the control based on wall displacement.
