In this paper, we study the boundary penalty method for optimal control of unsteady
Navier–Stokes type system that has been proposed as an alternative for Dirichlet boundary
control. Existence and uniqueness of solutions are demonstrated and existence of optimal
control for a class of optimal control problems is established. The asymptotic behavior of
solution, with respect to the penalty parameter ϵ, is studied. In particular, we prove convergence
of solutions of penalized control problem to the corresponding solutions of the Dirichlet
control problem, as the penalty parameter goes to zero. We also derive an optimality
system and determine optimal solutions. In order to illustrate the theoretical results and
the practical utility of control, we numerically address the problem of controlling
unsteady convection with Soret effect using a gradient-based method. Numerical results
show the effectiveness of the approach.