Exact controllability of an aeroacoustic model with a Neumann and a Dirichlet boundary control
Résumé
We study the exact controllability of a fluid-structure model. The fluctuations of fluid velocity and pressure in a domain are described by a potential , and the structure is a membrane located in a part of the boundary of the domain . The potential and the transverse displacement z of the membrane satisfy a coupled system of two wave equations, one in the domain and the other one in the boundary . We take two boundary controls, the first in a Neumann boundary condition on
satisfied by the potential and the second one in a Dirichlet boundary condition of the structure equation. We show that we can drive the fluid-structure system from an initial state in some space Y to another terminal state in Y with controls in . As in the case of the so-called Helmholtz fluid-structure model [J. Raymond and M. Vannination, ESAIM Control Optim. Calc. Var., 11 (2005), pp. 180–203] and in the aeroacoustic model with Dirichlet boundary controls [L. Cot, J.-P. Raymond, and J. Vancostenoble, Exact controllability of an aeroacoustic model, in CSVAA 2004—Control Set-Valued Analysis and Applications, EDP Sci., Les Ulis, 2007, pp. 26–49], the difficulty in the treatment of the observability inequalities, in the definition of very weak solutions, and in the proof of the controllability result comes from the coupling terms of the system. We show that the variants of the classical Hilbert uniqueness method introduced in [J. Raymond and M. Vannination, ESAIM Control Optim. Calc. Var., 11 (2005), pp. 180–203] and [L. Cot, J.-P. Raymond, and J. Vancostenoble, Exact controllability of an aeroacoustic model, in CSVAA 2004—Control Set-Valued Analysis and Applications, EDP Sci., Les Ulis, 2007, pp. 26–49] can be adapted to the aeroacoustic model that we consider.
