Time Reversal Focusing of the Initial State for Kirchhoff Plate

Abstract. Consider a Kirchhoff plate ∂2 t u+Δ 2u−∂2 t Δu = 0 in Ω× (0, T ), with boundary data u = Δu = 0 on ∂Ω×(0, T ) and unknown initial data u(·, 0) = u0 and ∂tu(·, 0) = u1 in Ω. We study an inverse problem of determining (u0, u1) from an interior observation u|ω×(0,T ). Here Ω is a bounded domain, ω a nonempty open subset of Ω, and T > 0 a suitable time duration. By means of an iterative time reversal technique, we derive an asymptotic formula of reconstructing (u0, u1) approximately with a logarithmical convergence rate for smooth initial data. The convergence becomes uniform and exponential when (Ω, ω, T ) satisfies the geometric control condition introduced by Bardos, Lebeau, and Rauch.