Decay rates at low and high frequencies for a plate equation with feedback concentrated in interior curves (Q2505202)

The paper deals the initial-boundary value problem for a plate equation with moment damping concentrated in interior curves \(u''-\Delta u(x,y,t)+\Delta^2 u(x,y,t) - \left(\int_\gamma \frac {\partial u'}{\partial \nu}d\gamma\right)\frac {\partial \delta_\gamma}{\partial \nu}=0,\;(x,y,t)\in Q,\) \(u(x,y,t)= \Delta u(x,y,t)=0,\;(x,y,t)\in \Sigma,\) \(u(x,y,0)=u_0(x,y),\;u'(x,y,0)=u^1(x,y),\;(x,y)\in \Omega\) with \(\Omega=(0,1)\times (0,1),\;\Gamma=\partial \Omega,\;Q=\Omega\times (0,+\infty),\;\Sigma=\Gamma \times (0,+\infty), \;\gamma=\partial((a,b)\times (a,b)),\;0<a<b<1,\;\delta_\gamma\) -- a Dirac mass concentrated in \(\gamma\). The main results concern the precise asymptotic behaviour of a solution. The exponential stability is proved for low frequencies but not for high frequencies. The explicit decay rate is derived for regular initial data at high frequencies. The method used is based on some observability inequalities for the corresponding undamped problem. It is shown numerically that the optimal location of the actuator is the the center of the domain \(\Omega.\)