Exact controllability of an aeroacoustic model

¹ Laboratoire MIP, UMR CNRS 5640, Université Paul Sabatier, 31062 Toulouse Cedex 4, France, cot@mip.ups-tlse.fr
² Laboratoire MIP, UMR CNRS 5640, Université Paul Sabatier, 31062 Toulouse Cedex 4, France, raymond@mip.ups-tlse.fr
³ Laboratoire MIP, UMR CNRS 5640, Université Paul Sabatier, 31062 Toulouse Cedex 4, France, vancoste@mip.ups-tlse.fr

Abstract

We study the exact controllability of a fluid-structure model. The fluctuations of velocity and pressure in the fluid are described by a potential, and the structure is a membrane located in a part of the boundary of the domain Ω. The potential ϕ and the transverse displacement z satisfy a coupled system of two wave equations, one in the domain , the other one in the boundary . Taking two boundary controls, the first one in a boundary condition satisfied by the potential, and the second one in a boundary condition of the structure equation, we identify the space of controllable initial conditions when the geometrical controllability conditions are satisfied. As in the case of the so-called Helmholtz fluid-structure model [10], the difficulty in the treatement of the observability inequalities, in the definition of very weak solutions, and in the proof of controllability result, comes from the coupling terms of the system. To overcome these difficulties, we show that the variant introduced in [10] of the classical Hilbert Uniqueness Method can be adapted to the aeroacoustic model we consider.