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A Kalman rank condition for the indirect controllability of coupled systems of linear operator groups

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Abstract

In this article, we give a necessary and sufficient condition of Kalman type for the indirect controllability of systems of groups of linear operators, under some “regularity and locality” conditions on the control operator that will be made precise later and fit very well the case of distributed controls. Moreover, in the case of first order in time systems, when the Kalman rank condition is not satisfied, we characterize exactly the initial conditions that can be controlled. Some applications to the control of systems of Schrödinger or wave equations are provided. The main tool used here is the fictitious control method coupled with the proof of an algebraic solvability property for some related underdetermined system and some regularity results.

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Acknowledgements

The authors would like to thank Camille Laurent for interesting discussions concerning Sect. 4 of this work.

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Authors and Affiliations

  1. Laboratoire Jacques-Louis Lions, CNRS UMR 7598, Université Pierre et Marie Curie (Univ. Paris 6), 75005, Paris, France

    Thibault Liard

  2. CEREMADE, Université Paris-Dauphine & CNRS UMR 7534, PSL Research University, 75016, Paris, France

    Pierre Lissy

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  1. Thibault Liard
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  2. Pierre Lissy
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Correspondence to Pierre Lissy.

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Funding

Pierre Lissy is partially supported by the project IFSMACS funded by the French Agence Nationale de la Recherche, 2015–2019 (Reference: ANR-15-CE40-0010).

Conflict of interest

The authors declare that they have no conflict of interest.

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Liard, T., Lissy, P. A Kalman rank condition for the indirect controllability of coupled systems of linear operator groups. Math. Control Signals Syst. 29, 9 (2017). https://doi.org/10.1007/s00498-017-0193-x

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  • DOI: https://doi.org/10.1007/s00498-017-0193-x

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