Identification problem for the wave equation with Neumann data input and Dirichlet data observations
The identification of the dispersive coefficient \(h(x)\in L^\infty\) in the wave equation in a bounded domain \(\Omega\) with \(C^2\) boundary \[ \begin{gathered} u_{tt}-\Delta u= h(x)u+ f(x,t),\quad (x,t)\in \Omega\times (0,T),\quad f\in L^2,\\ u(x,0)= u_0\in H^1(\Omega),\quad u_t(x,0)= u_1\in L^2(\Omega),\quad x\in\Omega,\\ {\partial u\over\partial n}= g\in H^{1/2}(\partial\Omega\times (0,T))\end{gathered} \] 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 \(0\leq t_1< t_2\leq T\), 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.
scientific article; zbMATH DE number 1880125
scientific article