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In this paper, we study an optimal control problem for the three-dimensional Navier-Stokes- α  equations in bounded domains with Dirichlet boundary conditions, where the time needed to reach a desired state plays an essential role. We first prove the existence of optimal solutions. Then we establish the first-order and second-order necessary optimality conditions, and the second-order sufficient optimality conditions. The second-order optimality ones obtained in the paper seem to be optimal in the sense that the gap between them is minimal.

Time Optimal Control Problem of the 3D Navier-Stokes- α Equations