Local null controllability of a two-dimensional fluid-structure interaction problem

Abstract

Abstract.In this paper, we prove a controllability result for a fluid-structure interaction problem.In dimension two, a rigid structure moves into an incompressible fluid governed by Navier-Stokes equations. The control acts on a fixed subset of the fluid domain. We prove that, for small initial data, this system is null controllable, that is, for a given T > 0, the system can be driven at rest and the structure to its reference configuration at time T . To show this result, we first consider a linearized system. Thanks to an observability inequality obtained from a Carleman inequality, we prove an optimal controllability result with a regular control. Next, with the help of Kakutani's fixed point theorem and a regularity result, we pass to the nonlinear problem.Mathematics Subject Classification. 35Q30, 93C20.

On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations

“…The results on semi-linear equations have been extended to the case of non smooth coefficients [DOP02,BDL07a,Le 07,LR10b]. The use of global parabolic Carleman estimates has also allowed to address the controllability of non linear equations such as the Navier-Stokes equations [Ima01,FCGIP04], the Boussinesq system [FCGIP06], fluid structure systems [IT07,BO08], weakly coupled parabolic systems [de 00, ABDK05, ABD06, GBPG06] to cite a few. A review of the application of global parabolic Carleman estimates can be found in [FCG06].…”

Feedback stabilization of a simplified 1d fluid–particle system

“…V S ≡ 0, this condition is trivially satisfied for all σ > 0. Let us also note that in dimension 2 or 3, for a rigid body immersed into a viscous incompressible fluid (and for a fluid domain that is connected), this problem disappear both for the controllability problem (see [20], [6], [5]) and for the stabilization problem (work in progress).…”

“…In Section 2 we first discuss the existence of solutions for (4) and on condition (6). In Section 3, we construct a change of variables to reduce the problem (1)- (3) to a cylindrical domain.…”

Null controllability in a fluid–solid structure model

“…Very recently Imanuvilov and Takahashi [12] and Boulakia and Osses [1] have studied the null controllability, locally about zero, of a system coupling the Navier-Stokes equations with the motion of a rigid body. Their analysis is based on Carleman estimates for a linearized system.…”

“…In the system coupling the Navier-Stokes equations with a rigid body, the domain occupied by the fluid depends on the position of the solid and therefore depends on the time variable. The linearized system may be stated either in a time dependent domain as in [1] or in a fixed domain as in [12]. In the present paper, we are going to establish Carleman inequalities for a linearized fluid-solid structure model, stated in a fixed domain Ω.…”