Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-29T09:23:58.489Z Has data issue: false hasContentIssue false
  • English
  • Français

Identification of a wave equation generated by a string∗

Published online by Cambridge University Press:  08 August 2014

Amin Boumenir
Amin Boumenir*
Affiliation:
Department of Mathematics, College of Science, Kuwait. University, P.O. Box 5969, 13060 Safat, Kuwait. boumenir@sci.kuniv.edu.kw
Get access

Abstract

We show that we can reconstruct two coefficients of a wave equation by a single boundary measurement of the solution. The identification and reconstruction are based on Krein’s inverse spectral theory for the first coefficient and on the Gelfand−Levitan theory for the second. To do so we use spectral estimation to extract the first spectrum and then interpolation to map the second one. The control of the solution is also studied.

Keywords

Inverse spectral methods; Krein string; Gelfand-Levitan theory
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 4 , October 2014 , pp. 1203 - 1213
Copyright
© EDP Sciences, SMAI, 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

L.E. Andersson, Algorithms for solving inverse eigenvalue problems for Sturm−Liouville equations, Inverse Methods in Action edited by P.C. Sabatier. Springer-Verlag (1990) 138–145.
Al-musallam, F. and Boumenir, A., Identification and control of a heat equation. Int. J. Evolution Equations 6 (2013) 85–100. Google Scholar
Avdonin, S.A. and Bulanova, A., Boundary control approach to the spectral estimation problem: the case of multiple poles. Math. Control Signals Systems 22 (2011) 245–265. Google Scholar
S.A. Avdonin, F. Gesztesy and A. Makarov, Spectral estimation and inverse initial boundary value problems. Vol. 4 of Inverse Probl. Imaging (2010) 1–9.
Barcilon, V., Iterative solution of the inverse Sturm−Liouville problem. J. Math. Phys. 15 (1974) 429–436. Google Scholar
Boumenir, A., The recovery of analytic potentials. Inverse Probl. 15 (1999) 1405–1423. Google Scholar
Boumenir, A., The reconstruction of an analytic string from two spectra. Inverse Probl. 20 (2004) 833–846. Google Scholar
Boumenir, A., Computing Eigenvalues of periodic Sturm−Liouville problems by the Shannon-Whittaker sampling theorem. Math. Comput. 68 (1999) 1057–1066. Google Scholar
Boumenir, A. and Tuan, Vu Kim, Recovery of the heat coefficient by two measurements. Inverse Probl. Imaging 5 (2011) 775–791 Google Scholar
Boumenir, A. and Tuan, Vu Kim, An inverse problem for the wave equation. J. Inverse Ill-Posed Probl. 19 (2011) 573–592. Google Scholar
Boumenir, A. and Zayed, A.I., Sampling with a String. J. Fourier Anal. Appl. 8 (2002) 211–232. Google Scholar
Boumenir, A. and Carroll, R., Toward a general theory of transmutation. Nonlin. Anal. 26 (1996) 1923–1936. Google Scholar
Carroll, R., Santosa, F., On the complete recovery of geophysical data. Math. Methods Appl. Sci. 4 (1982) 33–73. Google Scholar
H. Dym and H.P. McKean, Gaussian processes, function theory, and the inverse spectral problem. Vol. 31 of Probab. Math. Stat. Academic Press, New York, London (1976).
G.M.L. Gladwell, Inverse Problems in Vibration, series: Solid Mechanics and Its Applications, 2nd edition. Springer (2004).
Hald, O.H., The inverse Sturm−Liouville problem with symmetric potentials. Acta Math. 141 (1978) 263–291. Google Scholar
Hansen, S. and Zuazua, E., Exact controllability and stabilization of a vibrating string with an interior point mass. SIAM J. Control Optim. 33 (1995) 1357–1391. Google Scholar
S.I. Kabanikhin, A. Satybaev and M. Shishlenin, Direct Methods of Solving Multidimensional Inverse Hyperbolic Problems, Series: Inverse and Ill-Posed Problems Series. De Gruyter 48 (2005).
Kac, I.S. and Krein, M.G., On the spectral functions of the String. Amer. Math. Soc. Transl. 103 (1974) 19–102. Google Scholar
V. Komornik and P. Loreti, Fourier Series in Control Theory. Springer Monogr. Math. Springer (2005)
Küchler, U., K. Neumann An extension of Krein’s inverse spectral theorem to strings with non-reflecting left boundaries. Lect. Notes Math. 1485 (1991) 354–373. Google Scholar
Levitan, B.M. and Gasymov, M.G., Determination of a differential equation by two spectra. Russian Math. Surv. 19 (1964) 3–63. Google Scholar
McLaughlin, J.R., Stability theorems for two inverse spectral problems. Inverse Probl. 4 (1988) 529–540. Google Scholar
V. Marchenko, Sturm−Liouville Operators and Applications. Oper. Theory Adv. Appl., vol. 22. Birkhäuser, Basel (1986).
Privat, Y., Trélat, E. and Zuazua, E., Optimal Observation of the One-dimensional Wave Equation. J. Fourier Anal. Appl. 19 (2013) 514–544. Google Scholar
Rundell, W. and Sacks, P.E., Reconstruction techniques for classical inverse Sturm−Liouville problem. Math. Comput. 58 (1992) 161–183. Google Scholar
Seidman, T.I., Avdonin, S.A. and Ivanov, S.A., The ‘window problem’ for series of complex exponentials. J. Fourier Anal. Appl. 6 (2000) 233–254. Google Scholar
Sini, M., On the one-dimensional Gelfand and Borg−Levinson spectral problems for discontinuous coefficients. Inverse Probl. 20 (2004) 1371–1386. Google Scholar
Sini, M., Some uniqueness results of discontinuous coefficients for the one-dimensional inverse spectral problem. Inverse Probl. 19 (2003) 871–894. Google Scholar
Turchetti, G. and Sagretti, G., Stieltjes Functions and Approximation Solutions of an Inverse Problem. Springer Lect. Notes Phys. 85 (1978) 123-33. Google Scholar
V.S. Valdimirov, Equations of Mathematical Physics. Marcel Dekker, New York (1971).
A. Zayed, Advances in Shannon’s Sampling Theory. CRC Press, Boca Raton, FL (1993).