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

TL;DR

This paper proves that a deformable solid can stabilize a coupled fluid-solid system to zero velocity through boundary deformation control, with exponential decay, in 2D and 3D Navier-Stokes fluid models.

Contribution

It introduces a boundary feedback control method based on internal deformation of a self-propelled solid to stabilize fluid-solid interactions.

Findings

Velocities are stabilizable to zero with exponential decay.

Boundary deformation control can be derived from internal deformation.

Applicable to 2D and 3D Navier-Stokes fluid-solid systems.

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.