Time reversal focusing of the initial state for Kirchhoff plate

SIAM Journal on Applied Mathematics

Volumn 68, Issue 6, 2008, Pages 1535-1556

  Phung, Kim Dang ^a   Zhang, Xu ^a,b  

^a SICHUAN UNIVERSITY   (China)

^b INSTITUTE OF GEOLOGY AND GEOPHYSICS   (China)

Author keywords

Inverse problem; Kirchhoff plate; Observability estimate; Quantitative unique continuation; Time reversal technique

Indexed keywords

ASYMPTOTIC FORMULA; BOUNDARY DATA; BOUNDED DOMAIN; CONVERGENCE RATES; GEOMETRIC CONTROL; INITIAL STATE; KIRCHHOFF PLATES; TIME DURATION; TIME REVERSAL TECHNIQUES; TIME-REVERSAL FOCUSING; UNIQUE CONTINUATION;

INVERSE PROBLEMS;

EID: 67649322408     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/070684823     Document Type: Article

References (27)

1. C. Bardos and M. Fink, Mathematical foundations of the time reversal mirror, Asymptot. Anal., 29 (2002), pp. 157-182.
2. C. Bardos, G. Lebeau, and J. Rauch, Un exemple d'utilisation des notions de propagation pour le contr̂ole et la stabilisation des probl̀emes hyperboliques, in Nonlinear Hyperbolic Equations in Applied Mathematics, Rend. Sem. Mat. Univ. Politec. Torino 1988, (Special Issue) (1989), pp. 11-32.
3. C. Bardos, G. Lebeau, and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), pp. 1024-1065. (Pubitemid 23572519)
4. N. Burq, Contr̂olabilit́e exacte de l'́equation des ondes dans des ouverts peu ŕeguliers, Asymp-tot. Anal., 14 (1997), pp. 157-191. (Pubitemid 127729970)
5. O. Dorn, Time-Reversal and the Adjoint Method with an Application in Telecommunication, preprint, http://www.arxiv.org/abs/math.OC/0412379, 2004.
6. M. Fink, Time reversal of ultrasonic fields-basic principles, IEEE Trans. Ultrasonics Ferroelectric and Frequency Control, 39 (1992), pp. 555-556.
7. M. Fink and C. Prada, Acoustic time reversal mirrors, Inverse Problems, 17 (2001), pp. R1-R38.
8. J.-P. Fouque, J. Garnier, and A. Nachbin, Time reversal for dispersive waves in random media, SIAM J. Appl. Math., 64 (2004), pp. 1810-1838.
9. X. Fu, A weighted identity for partial differential operators of second order and its applications, C. R. Math. Acad. Sci. Paris, 342 (2006), pp. 579-584. (Pubitemid 43467736)
10. X. Fu, J. Yong, and X. Zhang, Exact controllability for multidimensional semilinear hyperbolic equations, SIAM J. Control Optim., 46 (2007), pp. 1578-1614.
11. A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series 34, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.
12. B. L. G. Jonsson, M. Gustafsson, V. H. Weston, and M. V. de Hoop, Retrofocusing of acoustic wave fields by iterated time reversal, SIAM J. Appl. Math., 64 (2004), pp. 1954-1986. (Pubitemid 40044164)
13. G. Lebeau and L. Robbiano, Contr̂ole exacte de l'́equation de la chaleur, Comm. Partial Differential Equations, 20 (1995), pp. 335-356.
14. G. Lebeau and L. Robbiano, Stabilisation de l'́equation des ondes par le bord, Duke Math. J., 86 (1997), pp. 465-491. (Pubitemid 127450670)
15. J.-L. Lions, Contr̂olabilit́e exacte, perturbations et stabilisation de syst̀emes distribúes. Tome 1: Contr̂ olabilit́e exacte, Rech. Math. Appl. 8, Masson, Paris, 1988.
16. K. Liu, B. Rao, and X. Zhang, Stabilization of the wave equations with potential and indefinite damping, J. Math. Anal. Appl., 269 (2002), pp. 747-769.
17. L. Miller, Escape function conditions for the observation, control, and stabilization of the wave equation, SIAM J. Control Optim., 41 (2003), pp. 1554-1566.
18. L. Pan, K. L. Teo, and X. Zhang, State-observation problem for a class of semi-linear hyperbolic systems, Chinese J. Contemp. Math., 25 (2004), pp. 163-172.
19. G. Papanicolaou, L. Ryzhik, and K. Sølna, Statistical stability in time reversal, SIAM J. Appl. Math., 64 (2004), pp. 1133-1155.
20. L. Robbiano, Fonction de coût et contr̂ole des solutions des ́equations hyperboliques, Asymptot. Anal., 10 (1995), pp. 95-115.
21. G. Wang and L. Wang, The Carleman inequality and its application to periodic optimal control governed by semilinear parabolic differential equations, J. Optim. Theory Appl., 118 (2003), pp. 249-461.
22. Y. Wang, Global uniqueness and stability for an inverse plate problem, J. Optim. Theory Appl., 132 (2007), pp. 161-173.
23. M. Yamamoto, Stability, reconstruction formula and regularization for an inverse source hyperbolic problem by a control method, Inverse Problems, 11 (1995), pp. 481-496.
24. M. Yamamoto, Uniqueness and stability in multidimensional hyperbolic inverse problems, J. Math. Pures Appl., 78 (1999), pp. 65-98.
25. M. Yamamoto and X. Zhang, Global uniqueness and stability for an inverse wave source problem for less regular data, J. Math. Anal. Appl., 263 (2001), pp. 479-500. (Pubitemid 33132677)
26. M. Yamamoto and X. Zhang, Global uniqueness and stability for a class of multidimensional inverse hyperbolic problems with two unknowns, Appl. Math. Optim., 48 (2003), pp. 211-228.
27. X. Zhang and E. Zuazua, Long time behavior of a coupled heat-wave system arising in fluid-structure interaction, Arch. Ration. Mech. Anal., 184 (2007), pp. 49-120.