The exponential stability of a coupled hyperbolic/parabolic system arising in structural acoustics
The paper studies uniform stabilization of a coupled system of wave and beam equations. Let \(\Omega\) be a bounded domain in \(\mathbb{R}^n\), \(n\geq 2\), with smooth boundary \(\Gamma=\Gamma_0 \cup\Gamma_1\), \(\Gamma_i\neq \emptyset\). The wave equation with state \(z\) is then described by \[ \begin{cases} z_{tt}= \Delta z\quad &\text{in }(0,\infty)\times \Omega,\\ z(0,x)= z_0,\;z_t(0,x)=z_1 \quad & \text{in }\Omega, \\ z(t,x)=0 \quad & \text{on }(0, \infty) \times\Gamma_0, \\ z_\nu+\alpha z_t= v_t\quad & \text{on }(0,\infty) \times\Gamma_1, \text{ with }\alpha >0\end{cases} \] where \(v\) denotes another state occupying the beam equation in \((0,\infty) \times\Gamma_1\) with the homogeneous boundary condition \(v=v_\nu=0\) on \(\partial\Gamma_1\), and \(z_t\) enters the beam equation as a distributed input. The aim is to obtain an exponential decay estimate to the energy \(E(z,z_t,v,v_t,t)\): \[ E(z,z_t,v,v_t, t)= \int_\Omega\bigl[ |\nabla z|^2+ |z_t|^2 \bigr]d\Omega+ \int_{\Gamma_1} \bigl[|\Delta v|^2+ |v_t|^2 \bigr]d\Gamma_1,\;t\geq 0. \] By assuming a geometrical condition on \(\Omega\) weaker than a ``star-shaped'' one, the decay is derived via an energy inequality of the form: \(E(z,z_t,v,v_t,T) \leq\eta E(z,z_t,v,v_t,0)\) for some \(T>0\) and \(0<\eta<1\) and the strong damping effect provided by the beam equation.
exponential decay estimate to the energy
scientific article; zbMATH DE number 1370621