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Error estimates for optimal control problems involving the Stokes system and Dirac measures

The aim of this work is to derive a priori error estimates for finite element discretizations of control--constrained optimal control problems that involve the Stokes system and Dirac measures. The first problem entails the minimization of a cost functional that involves point evaluations of the velocity field that solves the state equations. This leads to an adjoint problem with a linear combination of Dirac measures as a forcing term and whose solution exhibits reduced regularity properties. The second problem involves a control variable that corresponds to the amplitude of forces modeled as point sources. This leads to a solution of the state equations with reduced regularity properties. For each problem, we propose a finite element solution technique and derive a priori error estimates. Finally, we present numerical experiments, in two and three dimensions, that illustrate our theoretical developments.