Local null controllability of a rigid body moving into a Boussinesq flow

Abstract

In this paper, we study the controllability of a fluid-structure interaction system. We consider a viscous and incompressible fluid modeled by the Boussinesq system and the structure is a rigid body with arbitrary shape which satisfies Newton's laws of motion. We assume that the motion of this system is bidimensional in space. We prove the local null controllability for the velocity and temperature of the fluid and for the position and velocity of rigid body for a control acting only on the temperature equation on a fixed subset of the fluid domain.

Approximate controllability and stabilizability of a linearized system for the interaction between a viscoelastic fluid and a rigid body

“…These results have been extended to 3D and for a general shaped rigid body in [3]. Finally, the authors in [30] prove the local null controllability for a 2D Boussinesq flow in interaction with a rigid body by using controls acting only on the temperature equation.…”

Local null controllability of a fluid–rigid body interaction problem with Navier slip boundary conditions

“…Taking (s, λ) large enough, the fifth term in the right hand side of (5.20) can be transported to the left side. Indeed, since ϕ S is rigid, from Lemma 2.2 of [27], we have…”

“…In dimension 3, we mention [6], the same result was proved without any assumptions on the solid geometry while a condition of smallness on the H 2 norm of the initial fluid velocity is needed. We also mention [27], where the authors considered the interaction between a viscous and incompressible fluid modeled by the Boussinesq system and a rigid body with arbitrary shape, they proved null controllability of the associated system. In the case of the stabilization of fluid-solid ineraction systems, we have [2,3].…”

“…Second, one can also consider a heat conducting fluid and remplace the Navier-Stokes system by the Boussinesq system. This has been done for instance in [27] with a rigid body and Dirichlet boundary conditions. Our method here should be adapted to this case and we would obtain a similar result.…”

Controllability of a simplified fluid-structure interaction system

“…We expect to extend some of the tools developed here to handle the control properties of the system (1.2) in future works. The controllability properties of fluid-structure interaction systems have been tackled mainly in the case where the structure is a rigid body (see, [42], [18], [31], [23], [9], [8], [45], [17], [16], etc.). In [36], the author shows an observability inequality for the adjoint of a linearized and simplified fluid-structure interaction system in the case of a compressible viscous fluid and of a damped beam.…”