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Zbl 1031.35148
Feng, Xiaobing; Lenhart, Suzanne; Protopopescu, Vladimir; Rachele, Lizabeth; Sutton, Brian
Identification problem for the wave equation with Neumann data input and Dirichlet data observations.
(English)
[J] Nonlinear Anal., Theory Methods Appl. 52, No.7, A, 1777-1795 (2003). ISSN 0362-546X

The identification of the dispersive coefficient $h(x)\in L^\infty$ in the wave equation in a bounded domain $\Omega$ with $C^2$ boundary $$\gather u_{tt}-\Delta u= h(x)u+ f(x,t),\quad (x,t)\in \Omega\times (0,T),\quad f\in L^2,\\ u(x,0)= u_0\in H^1(\Omega),\quad u_t(x,0)= u_1\in L^2(\Omega),\quad x\in\Omega,\\ {\partial u\over\partial n}= g\in H^{1/2}(\partial\Omega\times (0,T))\endgather$$ is obtained by minimizing the Tikhonov functional $$J_\beta(h):= {1\over 2} \Biggl(\int_{\partial\Omega\times (t_1,t_2)} (u(h)- z)^2 ds dt+ \beta \int_\Omega h^2 dx\Biggr),$$ over $h\in L^\infty(\Omega)$, where $z\in L^2(\partial\Omega\times (t_1,t_2))$ with $0\le t_1< t_2\le T$, is a given data for $u|_{\partial\Omega\times (t_1,t_2)}$. However, no criterion for choosing the regularization parameter $\beta> 0$ 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.
[D.Lesnic (Leeds)]
MSC 2000:
*35R30 Inverse problems for PDE
35L05 Wave equation
65M32 Inverse problems

Keywords: coefficient identification; wave equation; optimal control; Dirichlet-Neumann map; Tikhonov's regularization

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