Abstract: The velocity tracking problem for the evolutionary Navier-Stokes equations in 2d is studied. The controls are of distributed type but the cost functional does not involve the usual quadratic term for the control. As a consequence the resulting controls can be of bang-bang type. First and second order necessary and suffcient conditions are proved. A fully-discrete scheme based on discontinuous (in time) Galerkin approach combined with conforming finite element subspaces in space, is proposed and analyzed. Provided that the time and space discretization parameters, t and h respectively, satisfy t_> Ch2 , then L2 error estimates are proved for the difference between the states corresponding to locally optimal controls and their discrete approximations.

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