Boundary controllability of a coupled wave/Kirchoff system

We consider two problems in boundary controllability of coupled wave/Kirchoff systems. Let Ω be a bounded region in R, n ≥ 2, with Lipschitz continuous boundary Γ. In the motivating structural acoustics application, Ω represents an acoustic cavity. Let Γ0 be a flat subset of Γ which represents a flexible wall of the cavity. Let z denote the acoustic velocity potential, which satisfies a wave equation in Ω, and let v denote the displacement on Γ0, which satisfies a Kirchoff plate equation on Γ0. These equations are coupled via ∂z/∂ν = vt on Γ0 (where ν is the exterior unit normal to Γ0), and the backpressure −zt appears in the Kirchoff equation. In the first problem, we consider a control u0 in the Kirchoff equation on Γ0, and an additional control u1 in the Neumann conditions on a subset Γ1 of Γ, where Γ \ Γ1 satisfies geometric conditions. Using both controls, we obtain exact controllability of the wave and plate components in the natural state space. In the second problem we consider only the control u0. Without geometric conditions, exact controllability is not possible, but we show that for any initial data, we can steer the plate component exactly, and the wave component approximately. ∗This work was partially supported by NSF grant DMS-0196359 . †This work was partially supported by NSF grant DMS-0104305 and ARO grant DAAD19-02-1-0179. ‡This work was partially supported by NSF grant DMS-0206951.