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An Optimal Control Problem for the Navier–Stokes Equations with Point Sources

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Abstract

We analyze, in two dimensions, an optimal control problem for the Navier–Stokes equations where the control variable corresponds to the amplitude of forces modeled as point sources; control constraints are also considered. This particular setting leads to solutions to the state equation exhibiting reduced regularity properties. We operate under the framework of Muckenhoupt weights, Muckenhoupt-weighted Sobolev spaces, and the corresponding weighted norm inequalities and derive the existence of optimal solutions and first- and, necessary and sufficient, second-order optimality conditions.

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Acknowledgements

FF is supported by UTFSM through Beca de Mantención. FL is partially supported by DIUBB through project 2120173 GI/C Universidad del Bío–Bío and ANID through FONDECYT grant 11200529. EO is partially supported by ANID through FONDECYT grant 1220156. DQ is partially supported by ANID/Subdirección del Capital Humano/Doctorado Nacional/2021–21210988.

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Authors and Affiliations

  1. Universidad Técnica Federico Santa María, Valparaíso, Chile

    Francisco Fuica, Enrique Otárola & Daniel Quero

  2. Universidad del Bío Bío, Concepción, Chile

    Felipe Lepe

Authors
  1. Francisco Fuica
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  2. Felipe Lepe
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  3. Enrique Otárola
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  4. Daniel Quero
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Correspondence to Enrique Otárola.

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Communicated by Michael Hinze.

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Fuica, F., Lepe, F., Otárola, E. et al. An Optimal Control Problem for the Navier–Stokes Equations with Point Sources. J Optim Theory Appl 196, 590–616 (2023). https://doi.org/10.1007/s10957-022-02148-2

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  • DOI: https://doi.org/10.1007/s10957-022-02148-2

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