On some applications of the boundary control method to spectral estimation and inverse problems
https://doi.org/10.17586/222080542015616378
Abstract
We consider applications of the Boundary Control (BC) method to generalized spectral estimation problems and to inverse source problems. We derive the equations of the BC method for these problems and show that the solvability of these equations crucially depends on the controllability properties of the corresponding dynamical system and properties of the corresponding families of exponentials.
Keywords
spectral estimation problem,
boundary control method,
identification problem,
inverse problem,
Schr¨odinger equation,
hyperbolic system
Supporting agencies: Sergei Avdonin was supported by the NSF grant DMS 1411564; Alexander Mikhaylov was supported by NSh1771.2014.1, RFBR 140100306, IRSES (FP7PEOPLE2012IRSES Marie Curie Actions); Victor Mikhaylov was supported by RFBR 140100535, RFBR 140131388 and NIR SPbGU 11.38.263.2014.
About the Authors
S. A. Avdonin
Department of Mathematics and Statistics, University of Alaska
United States
United States
Fairbanks
A. S. Mikhaylov
St. Petersburg Department of V.A. Steklov Institute of Mathematics of the Russian Academy of Sciences; St. Petersburg State University, Faculty of Mathematics and Mechanics
Russian Federation
Russian Federation
St. Petersburg
V. S. Mikhaylov
St. Petersburg Department of V.A. Steklov Institute of Mathematics of the Russian Academy
of Sciences; St. Petersburg State University, Faculty of Physics
Russian Federation
Russian Federation
St. Petersburg
References
1. Avdonin S., Bulanova A. Boundary control approach to the spectral estimation problem. The case of multiple poles. Math. Contr. Sign. Syst., 22 (3), P. 245–265 (2011).
2. Avdonin S., Bulanova A., Nicolsky D.. Boundary control approach to the spectral estimation problem. The case of simple poles. Sampling Theory in Signal and Image Processing, 8 (3), P. 225–248 (2009).
3. Avdonin S.A., Belishev M.I. Boundary control and dynamical inverse problem for nonselfadjoint Sturm–Liouville operator. Control Cybernetics, 26, P. 429–440 (1996).
4. Avdonin S., Gesztesy F., Makarov K.A. Spectral estimation and inverse initial boundary value problems. Inverse Probl. Imaging, 4 (1), P. 1–9 (2010).
5. Avdonin S., Ivanov S. Families of exponentials. Cambridge University Press, Cambridge, 1995.
6. Avdonin S., Mikhaylov V. The boundary control approach to inverse spectral theory. Inverse Problems, 26 (4), P. 045009(19) (2010).
7. Avdonin S., Mikhaylov V. Inverse source problem for the 1D Schr¨odinger equation. Zap. Nauchn. Sem. POMI, 393, P. 5–11 (2011).
8. Avdonin S.A., Mikhaylov V.S. Spectral estimation problem in infinite dimensional spaces. Zap. Nauchn. Sem. POMI, 422, P. 5–18 (2014).
9. Avdonin S., Mikhaylov V., Ramdani K. Reconstructing the potential for the 1D Schr¨odinger equation from boundary measurements, IMA Journal of Mathematical Control and Information, 31 (1), P. 137–150 (2014).
10. Baudouin L., Puel J.P. Uniqueness and stability in an inverse problem for the Schr¨odinger equation. Inverse Problems, 18 P. 1537–1554 (2002).
11. Belishev . Dynamical systems with boundary control: models and characterization of inverse data. Inverse Problems, 17 (4), P. 659–682 (2001).
12. Belishev M. On a relation between data of dynamic and spectral inverse problems. J. Math. Sci. (N. Y.), 127 (6), P. 2353–2363 (2005).
13. Belishev M. Recent progress in the boundary control method. Inverse Problems, 23, (2007).
14. Belishev M.I., Mikhailov V.S. Unified approach to classical equations of inverse problem theory. Journal of Inverse and IllPosed Problems, 20 (4), P. 461–488 (2012).
15. Mikhaylov A.S., Mikhaylov V.S. Equations of the Boundary Control method for the inverse source problem. Zap. Nauch. Sem POMI, 409, P. 121–128 (2012).
16. Mercado A., Axel Osses A., Lionel R. Carleman inequalities and inverse problems for the Schr¨odinger equation. C. R. Acad. Sci. Paris, Ser. I, 346, P. 53–58 (2008).
17. Po¨schel J., Trubowitz E. Inverse spectral theory. Boston, MA, Academic Press Inc., 1987.
18. Tucsnak M., Weiss G. Observation and control for operator semigroups. Birkha¨user Advanced Texts: Basler Lehrbucher. Birkha¨user Verlag, Basel, 2009, 483 p.
19. Trooshin I., Yamamoto M. Riesz basis of root vectors of a nonsymmetric system of firstorder ordinary differential operators and application to inverse eigenvalue problems. Appl. Anal., 80, (1–2), P. 19–51 (2001).
20. Trooshin I., Yamamoto M. Identification problem for a onedimensional vibrating system. Math. Methods Appl. Sci., 28 (17), P. 2037–2059 (2005).
Review
For citations:
Avdonin S.A., Mikhaylov A.S., Mikhaylov V.S. On some applications of the boundary control method to spectral estimation and inverse problems. Nanosystems: Physics, Chemistry, Mathematics. 2015;6(1):63–78. https://doi.org/10.17586/222080542015616378
Views:
0
Email this article 