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Local Exact Controllability to the Trajectories of the Cahn–Hilliard Equation

Applied Mathematics & Optimization volume 82, pages 279–306 (2020)Cite this article

Abstract

In this paper we prove the local exact controllability to the trajectories of the Cahn–Hilliard equation, which is a nonlinear fourth-order parabolic equation, by means of a control supported on an interior open interval. To prove this result we derive a Carleman estimate that allows us to conclude, thanks to a duality argument, the null controllability of the linearized equation around a given solution. Then, we apply a local inversion theorem to extend the control result to the nonlinear equation.

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Acknowledgements

The author dedicates this work to his parents, Mario Guzmán and Mariela Meléndez, for their unlimited love and support. This study has been partially supported by Basal Project FB0008, FONDECYT 1140741 and PIIC UTFSM 2015.

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Authors and Affiliations

  1. Departamento de Matemática, Universidad Técnica Federico Santa María, Avenida España 1680, Casilla 110-V, Valparaiso, Chile

    Patricio Guzmán

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  1. Patricio Guzmán
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Correspondence to Patricio Guzmán.

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Guzmán, P. Local Exact Controllability to the Trajectories of the Cahn–Hilliard Equation. Appl Math Optim 82, 279–306 (2020). https://doi.org/10.1007/s00245-018-9500-2

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  • DOI: https://doi.org/10.1007/s00245-018-9500-2

Keywords

  • Cahn–Hilliard equation
  • Parabolic equation
  • Internal control
  • Null controllability
  • Carleman estimates

Mathematics Subject Classification

  • 35K35
  • 93B05
  • 93B07