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Stabilization and Control for the Biharmonic Schrödinger Equation

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Abstract

The main purpose of this paper is to show the global stabilization and exact controllability properties of a fourth order nonlinear Schrödinger system on a periodic domain \(\mathbb {T}\) with internal control supported on an arbitrary sub-domain of \(\mathbb {T}\). More precisely, by certain properties of propagation of compactness and regularity in Bourgain spaces, for the solutions of the associated linear system, we show that the system is globally exponentially stabilizable. This property together with the local exact controllability shows that fourth order nonlinear Schrödinger is globally exactly controllable.

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Acknowledgements

The authors wish to thank the referee for his/her valuable comments which improved this paper. Capistrano–Filho was supported by CNPq (Brazil) Grants 306475/2017-0, 408181/2018-4, CAPES-PRINT (Brazil) Grant 88881.311964/2018-01 and Qualis A - Propesq (UFPE). This work was carried out during visits of the first author to the Federal University of Alagoas and visits of the second author to the Federal University of Pernambuco. The authors would like to thank both Universities for its hospitality. Finally, the authors would like to thanks C. Kwak for valuable suggestions on the proof of the Strichartz estimate.

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Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal de Pernambuco (UFPE), Recife, PE, 50740-545, Brazil

    Roberto A. Capistrano–Filho

  2. Instituto de Matemática, Universidade Federal de Alagoas (UFAL), Maceió, AL, Brazil

    Márcio Cavalcante

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  1. Roberto A. Capistrano–Filho
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Correspondence to Roberto A. Capistrano–Filho.

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Capistrano–Filho, R.A., Cavalcante, M. Stabilization and Control for the Biharmonic Schrödinger Equation. Appl Math Optim 84, 103–144 (2021). https://doi.org/10.1007/s00245-019-09640-8

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