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Identification problem for the wave equation with Neumann data input and Dirichlet data observations. (English) Zbl 1031.35148

The identification of the dispersive coefficient h(x)L in the wave equation in a bounded domain Ω with C 2 boundary

u tt -Δu=h(x)u+f(x,t),(x,t)Ω×(0,T),fL 2 ,u(x,0)=u 0 H 1 (Ω),u t (x,0)=u 1 L 2 (Ω),xΩ,u n=gH 1/2 (Ω×(0,T))

is obtained by minimizing the Tikhonov functional

J β (h):=1 2 Ω×(t 1 ,t 2 ) (u(h)-z) 2 d s d t + β Ω h 2 d x,

over hL (Ω), where zL 2 (Ω×(t 1 ,t 2 )) with 0t 1 <t 2 T, is a given data for u| Ω×(t 1 ,t 2 ) . However, no criterion for choosing the regularization parameter β>0 is given. Furthermore, some of the numerically obtained results for h(x) are 50% out of the corresponding analytical solution, showing that a more accurate numerical method for solving the nonlinear control problem is needed in any future work.

Reviewer: D.Lesnic (Leeds)
35R30Inverse problems for PDE
35L05Wave equation (hyperbolic PDE)
65M32Inverse problems (IVP of PDE, numerical methods)