Boundary controllability of structural acoustic systems with variable coefficients and curved walls

Abstract

This paper studies a structural acoustic model consisting of an interior acoustic wave equation with variable coefficients and a coupled Kirchhoff plate equation with a curved middle surface. By the Riemannian geometry approach and the multiplier technique, we establish exact controllability of the hybrid system under verifiable assumptions on the geometry of the interior domain and the interface boundary with two controls: One is a Neumann boundary control exerted on the wave equation, and the other acts on the interior of the plate equation. Furthermore, if the control for the plate equation is active alone, we prove that the hybrid system with partial Robin boundary condition of the wave equation is exactly controllable with the plate component and approximately controllable with the wave component.

Appendix: Derivation of the curved plate model in system (1.1)

In this appendix, we show how to use the Principle of Virtual Work to derive the (uncoupled) equations of motion for the vertical displacement w of the specific plate model under study in this paper. Let a plate have a curved middle surface $\varGamma _0,$ which is a part of a surface M in ${\mathbb {R}}^3$ with the induced metric $\tilde{g}$ from the Euclidean metric of ${\mathbb {R}}^3$. Then, the curved plate is given by

$$\begin{aligned} \{\,x+t\,\vec {n}\,|\,x\in \varGamma _0,\,\,t\in (-h/2,h/2)\,\}, \end{aligned}$$

where $\vec {n}$ is the unit normal on $\varGamma _0$ and $h>0$ is the thickness of the plate along its normal. We assume that the external deformation along the normal of the middle surface is small, i.e., the strain tensor of the middle surface is zero. Then, the strain energy of the bending of the plate is [23]

$$\begin{aligned} V=\frac{\textsf {D}}{2}\int _{\varGamma _0} a(w,w)\mathrm{d}\varGamma _0, \end{aligned}$$

where $a(w,w)=(1-\mu )\langle D^2_{\tilde{g}}w,D^2_{\tilde{g}}w\rangle _{T^2_x}+\mu (\Delta _{\tilde{g}} w)^2$, $\gamma =\frac{h^2}{12}$, $\textsf {D}=\frac{Eh\gamma }{1-\mu ^2}$ is called the modulus of flexural rigidity, $E,\mu$ and $\rho$ denote Young’s modulus, Poisson’s ratio and mass density, respectively. The kinetic energy of the plate is assumed to be

$$\begin{aligned} K=\frac{\rho h}{2}\int _{\varGamma _0}\left( w^2_t+\gamma |\nabla _{\tilde{g}}w_t|_{\tilde{g}}^2\right) \mathrm{d}\varGamma _0. \end{aligned}$$

Besides, we assume that there is no external loading on the plate. Accordingly, the corresponding Lagrangian for the plate is

$$\begin{aligned} L=\int ^T_0(K-V)\mathrm{d}t. \end{aligned}$$

Consider a plate which is clamped along the edge $\partial \varGamma _0\times (-h/2,h/2)$, that is $w=\frac{\partial w}{\partial \nu _{\tilde{g}}}=0~\text{ on }~~\partial \varGamma _0\times (0,T).$ The usual calculus of variations (Principle of Virtual Work) for a stationary point of L yields

$$\begin{aligned} \left\{ \begin{array}{ll} \rho hw_{tt}-\rho h\gamma \Delta _{\tilde{g}} w_{tt}+\textsf {D}\left[ \Delta ^2_{\tilde{g}}w-(1-\mu )\delta (k\hbox {d}w)\right] = 0&{}\quad \hbox {on}~(0,T)\times \varGamma _0,\\ w=\frac{\partial w}{\partial \nu _{\tilde{g}}}=0&{}\quad \hbox {on}~(0,T)\times \partial \varGamma _0,\\ w(0)=w_0,w_t(0)=w_1. \\ \end{array} \right. \nonumber \\ \end{aligned}$$

(5.1)

Making the change $t\rightarrow t\sqrt{\frac{\textsf {D}}{\rho h}}$ in the time scale, we can derive the (uncoupled version of the) bending plate model in system (1.1).