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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 $h(x)\in L^\infty$ 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.
Reviewer: D.Lesnic (Leeds)

MSC:
35R30Inverse problems for PDE
35L05Wave equation (hyperbolic PDE)
65M32Inverse problems (IVP of PDE, numerical methods)
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References:
[1] Banks, H. T.; Kunish, K. K.: Estimation techniques for distributed parameter systems. (1989)
[2] Belishev, M. I.: Boundary control in reconstruction of manifolds and metrics (the BC method). Inverse problem 13, R1-R45 (1997)
[3] Evans, L. C.: Partial differential equations. (1998) · Zbl 0902.35002
[4] Isakov, V.: Inverse problems for partial differential equations. (1998) · Zbl 0908.35134
[5] Isakov, V.; Sun, Z.: Stability estimates for hyperbolic inverse problems with local boundary data. Inverse problem 8, 193-206 (1992) · Zbl 0754.35184
[6] Lasiecka, I.; Triggiani, R.: Sharp regularity theory for second order hyperbolic equations of Neumann type part I--L2 nonhomogeneous data. Ann. math. Pura app. 157, 285-367 (1990) · Zbl 0742.35015
[7] I. Lasiecka, R. Triggiani, Differential and algebraic Riccati equations with applications to boundary/point control problems: continuous theory and approximation theory, in: Lecture Notes in Control and Information Sciences, Vol. 164, Springer, New York, 1991. · Zbl 0754.93038
[8] Lenhart, S.; Bhat, M.: Application of distributed parameter control model in wildlife damage management. Math. models methods appl. Sci. 2, 423-439 (1993) · Zbl 0770.92023
[9] Lenhart, S.; Liang, M.; Protopopescu, V.: Optimal control of boundary habitat hostility for interacting species. Math. methods appl. Sci. 22, 1061-1077 (1999) · Zbl 0980.92039
[10] Lenhart, S.; Protopopescu, V.; Yong, J.: Optimal control of a reflection boundary coefficient in an acoustic wave equation. Appl. anal. 69, 179-194 (1998) · Zbl 0903.49003
[11] Lenhart, S.; Protopopescu, V.; Yong, J.: Identification of boundary shape and reflexivity in a wave equation by optimal control techniques. Integral and differential equations 13, 941-972 (2000) · Zbl 0974.49013
[12] Liang, M.: Bilinear optimal control for a wave equation. Math. models methods appl. Sci. 9, 45-68 (1999) · Zbl 0939.49016
[13] J.L. Lions, E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. I, Springer, Berlin 1972. · Zbl 0223.35039
[14] Nachman, A. I.: Reconstruction from boundary measurements. Ann. math. 128, No. 2, 531-576 (1988) · Zbl 0675.35084
[15] Puel, J. P.; Yamamoto, M.: On a global estimate in a linear inverse hyperbolic problem. Inverse problems 12, 995-1002 (1996) · Zbl 0862.35141
[16] Rakesh, Reconstruction for an inverse problem for the wave equation with constant velocity, Inverse Problem 6 (1990) 91--98. · Zbl 0712.35104
[17] Rakesh, W.W. Symes, Uniqueness for an inverse problem for the wave equation, Commun. Partial Differential Equations 13 (1988) 87--96. · Zbl 0667.35071
[18] Richtmyer, R. D.; Morton, K. W.: Difference methods for initial-value problems. (1967) · Zbl 0155.47502
[19] Stoer, J.; Bulirsch, R.: Introduction to numerical analysis. (1993) · Zbl 0771.65002
[20] Sun, Z.: On continuous dependence for an inverse initial boundary value problem for the wave equation. J. math. Anal. appl. 115, 188-204 (1990) · Zbl 0733.35107
[21] Tikhonov, A. N.; Arsenin, V. Y.: Solutions of ill-posed problems. (1977) · Zbl 0354.65028
[22] Yamamoto, M.: Stability, reconstruction formula, and regularization for an inverse source hyperbolic problem by a control method. Inverse problems 11, 481-496 (1995) · Zbl 0822.35154
[23] M. Yamamoto, M. Masahiro, On an inverse problem of determining source terms in Maxwell’s equations with a single measurement in: A.G. Ramm (Ed.), Inverse Problems, Tomography, and Image Processing, Plenum, New York, 1998, pp. 241--256. · Zbl 0910.35144