An optimal control problem for the Navier-Stokes-$\alpha$ system

In this paper we study a distributed optimal control problem for a three-dimensional Navier-Stokes-$\alpha$ model. We prove the solvability of the optimal control problem, and derive first-order optimality conditions by using a Lagrange multipliers Theorem. Finally, considering a velocity tracking control problem for the three-dimensional Navier-Stokes-$\alpha$ model, we analyze the relation of its optimality system to the corresponding one associated to the Navier-Stokes model by proving a convergence theorem, which establishes that, as the length scale $\alpha$ goes to zero, the optimality system of the three-dimensional Navier-Stokes-$\alpha$ model converges to the optimality system associated with the velocity tracking control problem of the Navier-Stokes equations.