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Approximate controllability and stabilizability of a linearized system for the interaction between a viscoelastic fluid and a rigid body

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Abstract

We study control properties of a linearized fluid–structure interaction system, where the structure is a rigid body and where the fluid is a viscoelastic material. We establish the approximate controllability and the exponential stabilizability for the velocities of the fluid and of the rigid body and for the position of the rigid body. In order to prove this, we prove a general result for this kind of systems that generalizes in particular the case without structure. The exponential stabilization of the system is obtained with a finite-dimensional feedback control acting only on the momentum equation on a subset of the fluid domain and up to some rate that depends on the coefficients of the system. We also show that as in the case without structure, the system is not exactly null-controllable in finite time.

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Acknowledgements

Debanjana Mitra acknowledges the support from an Inspire Faculty Fellowship, RD/0118-DSTIN40-001. Arnab Roy was supported by the Czech Science Foundation (GAČR) project GA19-04243S. The Institute of Mathematics, CAS, is supported by RVO:67985840. Takéo Takahashi was partially supported by the ANR research project IFSMACS (ANR-15-CE40-0010). The three authors were partially supported by the IFCAM project “Analysis, Control and Homogenization of Complex Systems”.

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Bombay, Powai, Maharashtra, 400076, India

    Debanjana Mitra

  2. Institute of Mathematics, Czech Academy of Sciences, Žitná 25, 11567, Praha 1, Czech Republic

    Arnab Roy

  3. CNRS, Inria, IECL, Université de Lorraine, 54000, Nancy, France

    Takéo Takahashi

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  1. Debanjana Mitra
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  2. Arnab Roy
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  3. Takéo Takahashi
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Correspondence to Takéo Takahashi.

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Mitra, D., Roy, A. & Takahashi, T. Approximate controllability and stabilizability of a linearized system for the interaction between a viscoelastic fluid and a rigid body. Math. Control Signals Syst. 33, 637–667 (2021). https://doi.org/10.1007/s00498-021-00295-x

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