  We consider an optimal control problem for the Navier-Stokes system with
Navier slip boundary conditions. We denote by $\alpha$ the friction coefficient
and we analyze the asymptotic behavior of such a problem as $\alpha\to \infty$.
More precisely, we prove that if we take an optimal control for each $\alpha$,
then there exists a sequence of optimal controls converging to an optimal
control of the same optimal control problem for the Navier-Stokes system with
the Dirichlet boundary condition. We also show the convergence of the
corresponding direct and adjoint states.
