Chinese Annals of Mathematics, Series B

, Volume 39, Issue 2, pp 281–296 | Cite as

Internal Controllability for Parabolic Systems Involving Analytic Non-local Terms

  • Pierre Lissy
  • Enrique Zuazua


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.


Parabolic systems Non-local potentials Analyticity Null controllability Kalman rank condition Spectral unique continuation 

2000 MR Subject Classification

35K40 93B05 93B07 


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Copyright information

© Fudan University and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.CeremadeUniversité Paris-Dauphine & CNRS UMR 7534, PSLParisFrance
  2. 2.DeustoTechUniversity of DeustoBilbao, Basque CountrySpain
  3. 3.Departamento de MatemáticasUniversidad Autónoma de MadridMadridSpain
  4. 4.Facultad IngenieríaUniversidad de DeustoBasque CountrySpain
  5. 5.UPMC Univ. Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis LionsSorbonne UniversitésParisFrance

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