×

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 \[ \begin{gathered} u_{tt}-\Delta u= h(x)u+ f(x,t),\quad (x,t)\in \Omega\times (0,T),\quad f\in L^2,\\ u(x,0)= u_0\in H^1(\Omega),\quad u_t(x,0)= u_1\in L^2(\Omega),\quad x\in\Omega,\\ {\partial u\over\partial n}= g\in H^{1/2}(\partial\Omega\times (0,T))\end{gathered} \] 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.
Reviewer: D.Lesnic (Leeds)

MSC:

35R30 Inverse problems for PDEs
35L05 Wave equation
65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Banks, H. T.; Kunish, K. K., Estimation Techniques for Distributed Parameter Systems (1989), Birkhäuser: Birkhäuser Boston · Zbl 0695.93020
[2] Belishev, M. I., Boundary control in reconstruction of manifolds and metrics (the BC method), Inverse Problem, 13, R1-R45 (1997) · Zbl 0990.35135
[3] Evans, L. C., Partial Differential Equations (1998), American Mathematical Society: American Mathematical Society Providence, RI
[4] Isakov, V., Inverse Problems for Partial Differential Equations (1998), Springer: Springer Berlin · 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 \(I—L_2\) nonhomogeneous data, Ann. Math. Pura App., CLVII, 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.; 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.; 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, 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.; 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.; 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), Interscience Publishers: Interscience Publishers New York · Zbl 0155.47502
[19] Stoer, J.; Bulirsch, R., Introduction to Numerical Analysis (1993), Springer: Springer New York · 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), Wiley: Wiley New York · 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.; 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.