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An Insensitizing Control Problem for the Ginzburg–Landau Equation

  • Maurício Cardoso SantosEmail author
  • Thiago Yukio Tanaka
Article
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Abstract

In this paper, we prove the existence of insensitizing controls for the nonlinear Ginzburg–Landau equation. Here, we have a partially unknown initial data, and the problem consists in finding controls such that a specific functional is insensitive for small perturbations of the initial data. In general, the problem of finding controls with this property is equivalent to prove a partial null controllability result for an optimality system of cascade type. The novelty here is that we consider functionals depending on the gradient of the state, which leads to a null controllability problem for a system with second-order coupling terms. To manage coupling terms of this order, we need a new Carleman estimate for the solutions of the corresponding adjoint system.

Keywords

Ginzburg–Landau equation Carleman estimates Insensitizing controls Null controllability 

Mathematics Subject Classification

93C20 93B05 93B07 93C41 35K40 

Notes

Acknowledgements

The authors would like to thank professor Diego A. Souza, from Universidade Federal de Pernambuco (Brazil), to have participated in many discussions concerning the problems presented here, the authors are in debt for his support. The authors also thank the reviewers of this paper, to have raised many interesting questions and comments which certainly improved this work.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsFederal University of Paraíba, UFPBJoão PessoaBrazil
  2. 2.Department of MathematicsFederal Rural University of Pernambuco, UFRPERecifeBrazil

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