Finite element approximation and computations of boundary optimal control problems for the Navier-Stokes flows through a channel with steps

We study a boundary optimal control problem of the fluid flow governed by the Navier-Stokes equations. The control problem is formulated with the flow through a channel with steps. The first-order optimality condition of the optimal control is derived. Finite element approximations of the solutions of the optimality sys-tem are defined and optimal error estimates are derived. Finally, we present some numerical results.