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Error Estimates for Optimal Control Problems Involving the Stokes System and Dirac Measures

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Abstract

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.

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  1. Departamento de Matemática, Universidad Técnica Federico Santa María, Av. España 1680, Valparaíso, Chile

    Francisco Fuica, Enrique Otárola & Daniel Quero

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  1. Francisco Fuica
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  2. Enrique Otárola
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Correspondence to Enrique Otárola.

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FF is supported by UTFSM through Beca de Mantención. EO is partially supported by CONICYT through FONDECYT Project 11180193. DQ is partially supported by UTFSM through Programa de Incentivos a la Iniciación Científica (PIIC)

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Fuica, F., Otárola, E. & Quero, D. Error Estimates for Optimal Control Problems Involving the Stokes System and Dirac Measures. Appl Math Optim 84, 1717–1750 (2021). https://doi.org/10.1007/s00245-020-09693-0

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