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Internal Controllability for Parabolic Systems Involving Analytic Non-local Terms

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Abstract

This paper deals with the problem of internal controllability of a system of heat equations posed on a bounded domain with Dirichlet boundary conditions and perturbed with analytic non-local coupling terms. Each component of the system may be controlled in a different subdomain. Assuming that the unperturbed system is controllable—a property that has been recently characterized in terms of a Kalman-like rank condition—the authors give a necessary and sufficient condition for the controllability of the coupled system under the form of a unique continuation property for the corresponding elliptic eigenvalue system. The proof relies on a compactness-uniqueness argument, which is quite unusual in the context of parabolic systems, previously developed for scalar parabolic equations. The general result is illustrated by two simple examples.

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Authors and Affiliations

  1. Ceremade, Université Paris-Dauphine & CNRS UMR 7534, PSL, 75016, Paris, France

    Pierre Lissy

  2. DeustoTech, University of Deusto, 48007, Bilbao, Basque Country, Spain

    Enrique Zuazua

  3. Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049, Madrid, Spain

    Enrique Zuazua

  4. Facultad Ingeniería, Universidad de Deusto, Avda. Universidades, 24, 48007, Basque Country, Spain

    Enrique Zuazua

  5. UPMC Univ. Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, Sorbonne Universités, F-75005, Paris, France

    Enrique Zuazua

Authors
  1. Pierre Lissy
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  2. Enrique Zuazua
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Correspondence to Pierre Lissy.

Additional information

This article is dedicated to Phillippe G. Ciarlet in the occasion of his 80th birthday, with gratitude and admiration for his mastery and continuous support. Merci Philippe!

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Lissy, P., Zuazua, E. Internal Controllability for Parabolic Systems Involving Analytic Non-local Terms. Chin. Ann. Math. Ser. B 39, 281–296 (2018). https://doi.org/10.1007/s11401-018-1064-6

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  • DOI: https://doi.org/10.1007/s11401-018-1064-6

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