Geodesic
Identifiability for Linearized Sine-Gordon Equation
J. Ha1 ; S. Gutman2
1 School of Liberal Arts, Korea University of Technology and Education Cheonan 330-708, South Korea
2 Department of Mathematics, University of Oklahoma Norman, Oklahoma 73019, USA
Mathematical modelling of natural phenomena, Tome 8 (2013) no. 1, pp. 106-121.

Voir la notice de l'article provenant de la source EDP Sciences

The paper presents theoretical and numerical results on the identifiability, i.e. the unique identification for the one-dimensional sine-Gordon equation. The identifiability for nonlinear sine-Gordon equation remains an open question. In this paper we establish the identifiability for a linearized sine-Gordon problem. Our method consists of a careful analysis of the Laplace and Fourier transforms of the observation of the system, conducted at a single point. Numerical results based on the best fit to data method confirm that the identification is unique for a wide choice of initial approximations for the sought test parameters. Numerical results compare the identification for the nonlinear and the linearized problems.
DOI : 10.1051/mmnp/20138107

J. Ha 1 ; S. Gutman 2

1 School of Liberal Arts, Korea University of Technology and Education Cheonan 330-708, South Korea
2 Department of Mathematics, University of Oklahoma Norman, Oklahoma 73019, USA 
@article{MMNP_2013_8_1_a6,
     author = {J. Ha and S. Gutman},
     title = {Identifiability for {Linearized} {Sine-Gordon} {Equation}},
     journal = {Mathematical modelling of natural phenomena},
     pages = {106--121},
     publisher = {mathdoc},
     volume = {8},
     number = {1},
     year = {2013},
     doi = {10.1051/mmnp/20138107},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20138107/}
}
TY  - JOUR
AU  - J. Ha
AU  - S. Gutman
TI  - Identifiability for Linearized Sine-Gordon Equation
JO  - Mathematical modelling of natural phenomena
PY  - 2013
SP  - 106
EP  - 121
VL  - 8
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20138107/
DO  - 10.1051/mmnp/20138107
LA  - en
ID  - MMNP_2013_8_1_a6
ER  - 
%0 Journal Article
%A J. Ha
%A S. Gutman
%T Identifiability for Linearized Sine-Gordon Equation
%J Mathematical modelling of natural phenomena
%D 2013
%P 106-121
%V 8
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20138107/
%R 10.1051/mmnp/20138107
%G en
%F MMNP_2013_8_1_a6
J. Ha; S. Gutman. Identifiability for Linearized Sine-Gordon Equation. Mathematical modelling of natural phenomena, Tome 8 (2013) no. 1, pp. 106-121. doi : 10.1051/mmnp/20138107. http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20138107/

[1] A. R. Bishop, K. Fesser, P. S. Lomdahl Physica D 1983 259 279

[2] N. Dunford, J. Schwartz. Linear operators, Part I. Wiley Sons, New York, NY. Reprint of the 1958 original, 1988.

[3] L. C. Evans. Partial differential equations. Graduate studies in Mathematics, vol. 19. American Mathematical Society, Providence, R.I, 1998.

[4] S. Gutman, J. Ha SIAM J. Control and Optimization 2007 694 713

[5] S. Gutman, J. Ha Math. Comp. Simul. 2009 2192 2210

[6] J. Ha, S. Gutman J. Korean Math. Soc. 2009 1105 1117

[7] J. Ha, S. Nakagiri J. Korean Math. Soc. 2006 931 950

[8] V. Isakov. Inverse problems for partial differential equations. Second edition. Applied Mathematical Sciences, vol. 127. Springer, New York, 2006.

[9] S. Kitamura, S. Nakagiri SIAM J. Control and Optimization 1977 785 802

[10] M. Levi. Beating modes in the Josephson junction. Chaos in Nonlinear Dynamical Systems, J. Chandra (Ed.) (1984), SIAM, Philadelphia.

[11] J. L. Lions. Optimal control of systems governed by partial differential equations. Springer-Verlag, New York, 1971.

[12] A. C. Metaxas, R. J. Meredith. Industrial microwave heating. Peter Peregrinus, London, 1993.

[13] S. Nakagiri Inverse Problems 1993 143 191

[14] R. Ortega, A. M. Robles-Perez J. Math. Anal. Appl. 1998 625 651

[15] A. Pierce SIAM J. Control and Optimization 1979 494 499

[16] W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery. Numerical Recepies in FORTRAN, 2nd ed. Cambridge University Press, Cambridge, 1992.

[17] N. F. Smyth, A. L. Worthy. Soliton evolution and radiation loss for the sine-Gordon equation. Physical Reviews E, (1999), 2330–2336.

[18] M. Taylor. Partial differential equations I. Basic theory. Second edition. Applied Mathematical Sciences, vol. 115. Springer, New York, 2011.

[19] R. Temam. Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd ed. Applied Mathematical Sciences, vol. 68, Springer-Verlag, 1997.

[20] K. Yosida. Functional Analysis, 6th ed. Springer-Verlag, 1980.

Cité par Sources :