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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 $h\left(x\right)\in {L}^{\infty }$ in the wave equation in a bounded domain ${\Omega }$ with ${C}^{2}$ boundary

$\begin{array}{c}{u}_{tt}-{\Delta }u=h\left(x\right)u+f\left(x,t\right),\phantom{\rule{1.em}{0ex}}\left(x,t\right)\in {\Omega }×\left(0,T\right),\phantom{\rule{1.em}{0ex}}f\in {L}^{2},\\ u\left(x,0\right)={u}_{0}\in {H}^{1}\left({\Omega }\right),\phantom{\rule{1.em}{0ex}}{u}_{t}\left(x,0\right)={u}_{1}\in {L}^{2}\left({\Omega }\right),\phantom{\rule{1.em}{0ex}}x\in {\Omega },\\ \frac{\partial u}{\partial n}=g\in {H}^{1/2}\left(\partial {\Omega }×\left(0,T\right)\right)\end{array}$

is obtained by minimizing the Tikhonov functional

${J}_{\beta }\left(h\right):=\frac{1}{2}\left({\int }_{\partial {\Omega }×\left({t}_{1},{t}_{2}\right)}{\left(u\left(h\right)-z\right)}^{2}dsdt+\beta {\int }_{{\Omega }}{h}^{2}dx\right),$

over $h\in {L}^{\infty }\left({\Omega }\right)$, where $z\in {L}^{2}\left(\partial {\Omega }×\left({t}_{1},{t}_{2}\right)\right)$ with $0\le {t}_{1}<{t}_{2}\le T$, is a given data for ${u|}_{\partial {\Omega }×\left({t}_{1},{t}_{2}\right)}$. However, no criterion for choosing the regularization parameter $\beta >0$ is given. Furthermore, some of the numerically obtained results for $h\left(x\right)$ 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)
##### MSC:
 35R30 Inverse problems for PDE 35L05 Wave equation (hyperbolic PDE) 65M32 Inverse problems (IVP of PDE, numerical methods)