# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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:
 35R30 Inverse problems for PDE 35L05 Wave equation (hyperbolic PDE) 65M32 Inverse problems (IVP of PDE, numerical methods)
Full Text:
##### 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