Time Reversal Focusing of the Initial State for Kirchhoff Plate

Consider a Kirchhoff plate $\partial_t^2 u + \Delta^2 u - \partial_t^2 \Delta u = 0$ in $\Omega\times (0,T)$, with boundary data $u=\Delta u=0$ on $\partial\Omega \times (0,T)$ and unknown initial data $u(\cdot,0) = u_0$ and $\partial_t u(\cdot,0) = u_1$ in $\Omega$. We study an inverse problem of determining $(u_0,u_1)$ from an interior observation $u|_{\omega\times(0,T)}$. Here $\Omega$ is a bounded domain, $\omega$ a nonempty open subset of $\Omega$, and $T>0$ a suitable time duration. By means of an iterative time reversal technique, we derive an asymptotic formula of reconstructing $(u_0,u_1)$ approximately with a logarithmical convergence rate for smooth initial data. The convergence becomes uniform and exponential when $(\Omega,\omega,T)$ satisfies the geometric control condition introduced by Bardos, Lebeau, and Rauch.