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A Kalman condition for the controllability of a coupled system of Stokes equations

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Abstract

We consider the controllability of a class of systems of n Stokes equations, coupled through terms of order zero and controlled by m distributed controls. Our main result states that such a system is null-controllable if and only if a Kalman type condition is satisfied. This generalizes the case of finite-dimensional systems and the case of systems of coupled linear heat equations. The proof of the main result relies on the use of the Kalman operator introduced in [1] and on a Carleman estimate for a cascade type system of Stokes equations. Using a fixed-point argument, we also obtain that if the Kalman condition is verified, then the corresponding system of Navier–Stokes equations is locally null-controllable.

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Authors and Affiliations

  1. Université de Lorraine, CNRS, Inria, IECL, F-54000, Nancy, France

    Takéo Takahashi

  2. Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, C.U., C. P. 04510, Mexico City, Mexico

    Luz de Teresa & Yingying Wu-Zhang

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  1. Takéo Takahashi
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  2. Luz de Teresa
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  3. Yingying Wu-Zhang
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Correspondence to Takéo Takahashi.

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T. Takahashi: The research was partially supported by the French National Research Agency (ANR), Project TRECOS, ANR-20-CE40-0009. Luz de Teresa: The research was partially supported by Conahcyt, project A1-S-17475 and UNAM-PAPIIT IN109522. Yingying Wu-Zhang: Supported by Conahcyt scholarship 849458 and UNAM-Programa de Apoyo a los Estudios del Posgrado.

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Takahashi, T., de Teresa, L. & Wu-Zhang, Y. A Kalman condition for the controllability of a coupled system of Stokes equations. J. Evol. Equ. 24, 4 (2024). https://doi.org/10.1007/s00028-023-00935-6

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