Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part I: The linearized system.

Sébastien Court

doi:10.3934/eect.2014.3.59

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2014, Volume 3, Issue 1: 59-82. Doi: 10.3934/eect.2014.3.59

This issue Previous Article Optimal control for stochastic heat equation with memory Next Article Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part II: The nonlinear system.

Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part I: The linearized system.

Sébastien Court^1,

1.
Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 9

Received: February 28, 2013

Revised: October 31, 2013

Published: February 2014

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Abstract

This paper is the first part of a work which consists in proving the stabilization to zero of a fluid-solid system, in dimension 2 and 3. The considered system couples a deformable solid and a viscous incompressible fluid which satisfies the incompressible Navier-Stokes equations. By deforming itself, the solid can interact with the environing fluid and then move itself. The control function represents nothing else than the deformation of the solid in its own frame of reference. We there prove that the velocities of the linearized system are stabilizable to zero with an arbitrary exponential decay rate, by a boundary deformation velocity which can be chosen in the form of a feedback operator. We then show that this boundary feedback operator can be obtained from an internal deformation of the solid which satisfies the linearized physical constraints that a self-propelled solid has to satisfy.

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Mathematics Subject Classification: Primary: 93C20, 35Q30, 76D05, 76D07, 74F10, 93C05, 93B52, 93D15; Secondary: 74A99, 35Q74.

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