Geodesic
Identification of boundary conditions at one of the ends of a segment
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 19 (2017) no. 3, pp. 11-23.

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We consider a boundary value problem on an interval for a fourth-order differential equation. The boundary conditions at one end of the segment are known, but at the other end of the segment they are unknown. The eigenvalues of the boundary value problem are known as well. The problem is to reconstruct the unknown boundary conditions at one of the ends of the segment. Four theorems are proved in the paper. The first two theorems are algebraic. They show that the matrix can be reconstructed accurate within linear transformations of rows with respect to its minors of maximal order. In this case, the matching conditions (so called Plucker relations) must be satisfied for the minors. In two other theorems, on the basis of the first two theorems, we prove the duality of the reconstruction of boundary conditions. The third theorem is devoted to the identification of boundary conditions by the entire spectrum of eigenvalues, and the fourth is to identify boundary conditions with respect to a finite number of eigenvalues. It is shown that it is sufficient to use four eigenvalues to identify the boundary conditions. Examples of the identification problem’s solution are given.
Keywords: boundary conditions, inverse problem, eigenvalues, differential equation of the fourth order, Plucker relations. 
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A. M. Akhtyamov; R. Y. Galimov; A. V. Mouftakhov. Identification of boundary conditions at one of the ends of a segment. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 19 (2017) no. 3, pp. 11-23. http://geodesic.mathdoc.fr/item/SVMO_2017_19_3_a0/

[1] G. M. L. Gradwell, Inverse Problems in Vibration, NIC Regulyarnaya i haoticheskaya dinamika, Institut komp'yuternyh iissledovanij, M.; Ishevsk, 2008, 608 pp. (In Russ.)

[2] M. T. Chu, G. H. Golub, Inverse Eigenvalue Problems: Theory, Algorithms, and Applications, University Press, Oxford, 2005, 406 pp. | MR | Zbl

[3] M. A. Naymark, Linear differential operators, Nauka, M., 1969, 526 pp. (In Russ.) | MR

[4] V. A. Marchenko, The operators of the Sturm-Liouville problem and their applications, Naukova dumka, Kiev, 1977, 332 pp. (In Russ.) | MR

[5] B. M. Levitan, The inverse Sturm-Liouville problem, Nauka, M., 1984, 240 pp. (In Russ.) | MR

[6] A. O. Vatul'yan, Inverse problems in solid mechanics, Fizmatlit, M., 2007, 224 pp. (In Russ.)

[7] A. O. Vatul'yan, A. V. Osipov, “About one approach to determine the parameters of the defect in the beam”, Defektoskopiya, 11 (2014), 37-47 (In Russ.)

[8] E. I. Shifrin, R. Ruotolo, “Natural frequencies of a beam with an arbitrary number of cracks”, Journal of Sound and Vibration, 222:3 (1999), 409–423 | DOI

[9] E. I. Shifrin, “Identification of a finite number of small cracks in a rod using natural frequencies”, Mechanical Systems and Signal Processing, 70–71 (2016), 613–624 | DOI

[10] M. A. Il'gamov, A. G. Hakimov, “Diagnosis of injuries of the cantilever beam with notch”, Defektoskopiya., 6 (2009), 83-89 (In Russ.)

[11] M. A. Il'gamov, A. G. Hakimov, “Diagnosis of fastening and damage beams on elastic supports”, Kontrol'. Defektoskopiya., 9 (2010), 57–63 (In Russ.)

[12] V. C. Gnuni, Z. B. Oganisyan, “The definition of boundary conditions of circular ring plates on the set frequencies of own fluctuations”, Izvestiya NAN RA, seriya “Mekhanika”., 44:5 (1991), 9-16 (In Russ.)

[13] Z. B. Oganisyan, “On one problem of reconstruction of boundary conditions on the ends of the rod at a predetermined range of frequencies of own fluctuations of cross”, Questions of optimal control, stability and strength of mechanical systems (scientific conference proceedings), Conference proceedings, Yerevan, 1997, 159-162 (In Russ.)

[14] Z. B. Oganisyan, “On one problem of reconstruction of boundary conditions on the edges of the plate at a given spectrum of frequencies of own fluctuations of cross”, Uchenye zapiski EGU., 1 (1991), 45-50 (In Russ.) | Zbl

[15] A. M. Akhtyamov, A. V. Mouftakhov, “Identification of boundary conditions using natural frequencies”, Inverse Problems in Science and Engineering, 12:4 (2004), 393–408 | DOI | MR

[16] A. M. Ahtyamov, A. V. Muftahov, “The correctness according to Tikhonov identification problem of fixing mechanical systems”, Sibirskij zhurnal industrial'noj matematiki., 25:4(52) (2012), 24-37 (In Russ.) | MR

[17] A. A. Aitbaeva, A. M. Ahtyamov, “On uniqueness of determination of the type of boundary conditions at one end of the rod at the three natural frequencies of oscillation”, Prikladnaya matematika i mekhanika., 80:3 (2016), 388-394 (In Russ.)

[18] A. V. Mouftakhov, On the Reconstruction of the Matrix from its Minors, 2006, arXiv: math/0603657

[19] P. Lankaster, Theory of matrices, Nauka, M., 1982, 272 pp. (In Russ.) | MR

[20] V. V. Bolotin, Vibration in engineering, Kolebaniya linejnyh sistem [Vibrations of linear systems], Mashinostroenie, M., 1978, 352 pp.