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Pré-Publication, Document De Travail Année : 2023

Controllability with one scalar control of a system of interaction between the Navier-Stokes system and a damped beam equation

Résumé

We consider the controllability of a fluid-structure interaction system, where the fluid is modeled by the Navier-Stokes system and where the structure is a damped beam located on a part of its boundary. The motion of the fluid is bi-dimensional whereas the deformation of the structure is one-dimensional and we use periodic boundary conditions in the horizontal direction. Our result is the local null-controllability of this free-boundary system by using only one scalar control acting on an arbitrary small part of the fluid domain. This improves a previous result obtained by the authors where three scalar controls were needed to achieve the local null-controllability. In order to show the result, we prove the final-state observability of a linear Stokes-beam interaction system in a cylindrical domain. This is done by using a Fourier decomposition, proving Carleman inequalities for the corresponding system for the low-frequencies solutions and in the case where the observation domain is an horizontal strip. Then we conclude this observability result by using a Lebeau-Robbiano strategy for the heat equation and a uniform exponential decay for the high-frequencies solutions. Then, the result on the nonlinear system can be obtained by a change of variables and a fixed-point argument.
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Dates et versions

hal-04140088 , version 1 (24-06-2023)

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  • HAL Id : hal-04140088 , version 1

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Rémi Buffe, Takéo Takahashi. Controllability with one scalar control of a system of interaction between the Navier-Stokes system and a damped beam equation. 2023. ⟨hal-04140088⟩
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