Abstract
We study the feedback stabilization of a system composed by an incompressible viscous fluid and a deformable structure located at the boundary of the fluid domain. We stabilize the position and the velocity of the structure and the velocity of the fluid around a stationary state by means of a Dirichlet control, localized on the exterior boundary of the fluid domain and with values in a finite dimensional space. Our result concerns weak solutions for initial data close to the stationary state. Our method is based on general arguments for stabilization of nonlinear parabolic systems combined with a change of variables to handle the fact that the fluid domain of the stationary state and of the stabilized solution are different. We prove that for initial data close to the stationary state, we can stabilize the position and the velocity of the deformable structure and the velocity of the fluid.
Highlights
We consider the problem of stabilization for a fluid-structure system composed by a viscous incompressible fluid and a deformable structure located at the boundary of the fluid domain
Our aim consists in showing the boundary stabilization of such a system in the 2d case and for weak solutions
In the 3d case or for strong solutions, the stabilization of such fluid-structure systems could be obtained by using the methodology developed in [6] or in [8]
Summary
We consider the problem of stabilization for a fluid-structure system composed by a viscous incompressible fluid and a deformable structure located at the boundary of the fluid domain. In order to do this here, a first step consists in performing a change of variables to work on a cylindrical domain (see Section 2) Such an approach is already considered for strong solutions and there exists changes of variables that allow to keep the divergence free conditions and the form of the boundary conditions. That is classical in fluid-structure interaction problems, is coming from the fact that the solutions and the test functions of (1.18) and of (1.19) are not written in the same spatial domains To overcome this issue, we transform the system (1.18) by using a change of variables X such that X(t, F (ηS)) = F (η(t, ·)) and X(t, Γstr(ηS)) = Γstr(η(t, ·)), and such that X(t, ·) = Id on Γ0 for all t. We postpone technical proofs to the three sections: Section A is devoted to the change of variables, Section B to the linearization, and Section C to some estimates for the fixed point
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