Global Uniqueness and Stability for a Class of Multidimensional Inverse Hyperbolic Problems with Two Unknowns

Applied Mathematics and Optimization

Volumn 48, Issue 3, 2003, Pages 211-228

  Yamamoto, Masahiro ^a   Zhang, Xu ^b,c  

^a UNIVERSITY OF TOKYO   (Japan)

^b SICHUAN UNIVERSITY   (China)

^c UNIVERSIDAD AUTÓNOMA DE MADRID   (Spain)

Author keywords

Global uniqueness; Inverse hyperbolic problem; Observability; Stability estimate; Two unknowns

Indexed keywords

BOUNDARY VALUE PROBLEMS; DIFFERENTIAL EQUATIONS; FUNCTION EVALUATION; INTEGRATION; MAPPING; PROBLEM SOLVING; SET THEORY; STABILITY;

GLOBAL UNIQUENESS; INVERSE HYPERBOLIC PROBLEMS;

INVERSE PROBLEMS;

EID: 0348013281     PISSN: 00954616     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00245-003-0775-5     Document Type: Article

References (26)

1. 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), 1024-1065.
2. L. Hörmander, The Analysis of Linear Partial Differential Operators, III, Pseudodifferential Operators, Springer-Verlag, Berlin, 1985.
3. O. Yu. Imanuvilov, On Carleman estimates for hyperbolic equations, Asymp. Anal., 32 (2002), 185-220.
4. V. Isakov. Inverse Source Problems, Mathematical Surveys and Monographs, vol. 34, American Mathematical Society, Providence, RI, 1990.
5. V. Isakov, Inverse Problems for Partial Differential Equations, Springer-Verlag, Berlin, 1998.
6. J. U. Kim, Control of a plate equation with large memory, Differential Integral Equations, 5 (1992), 261-279.
7. J. U. Kim, Control of a second-order integro-differential equation, SIAM J. Control Optim., 31 (1993), 101-110.
8. M. V. Klibanov, Carleman estimates and inverse problems in the last two decades, in Surveys on Solutions Methods for Inverse Problems Springer-Verlag, Wien, 2000, pp. 119-146.
9. V. Komornik, Exact Controllability and Stabilization, the Multiplier Method, Masson, Paris, 1994.
10. I. Lasiecka, Controllability of a viscoelastic Kirchhoff plate, in Control and Estimation of Distributed Parameter Systems, International Series of Numerical Mathematics, vol. 91, Birkhäuser, Basel, 1989, pp. 237-247.
11. I. Lasiecka, J. L. Lions and R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl., 65 (1986), 149-192.
12. I. Lasiecka and R. Triggiani, Exact controllability of semilinear abstract systems with application to waves and plates boundary control problems, Appl. Math. Optim., 23 (1991), 109-154.
13. I. Lasiecka and R. Triggiani, Carleman estimates and exact boundary controllability for a system of coupled, nonconservative second-order hyperbolic equations, in Partial Differential Equation Methods in Control and Shape Analysis (Pisa), Lecture Notes in Pure and Applied Mathematics, 188, Dekker, New York, 1997, pp. 215-243.
14. I. Lasiecka, R. Triggiani and X. Zhang, Nonconservative wave equations with unobserved Neumann B.C.: global uniqueness and observability in one shot, in Differential Geometric Methods in the Control of Partial Differential Equations, Contemporary Mathematics, vol. 268, American Mathematical Society, Providence, RI, 2000, pp. 227-325.
15. M. M. Lavrent'ev, V. G. Romanov and S. P. Shishatskii, Ill-posed problems of mathematical physics and analysis, translated from the Russian by J. R. Schulenberger, Translations of Mathematical Monographs, vol. 64, American Mathematical Society, Providence, RI, 1986.
16. G. Leugering, Exact boundary controllability of an integro-differential equation, Appl. Math. Optim., 15 (1987), 223-250.
17. J. L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués, Tome 1, Masson, Paris, 1988.
18. K. Liu, M. Yamamoto and X. Zhang, Observability inequalities by internal observation and its applications, J. Optim. Theory Appl., 116 (2003), 621-645.
19. A. López, X. Zhang and E. Zuazua, Null controllability of the heat equation as singular limit of the exact controllability of dissipative wave equations, J. Math. Pures Appl., 79 (2000), 741-808.
20. J.-P. Puel and M. Yamamoto, On a global estimate in a linear inverse hyperbolic problem, Inverse Problems, 12 (1996), 995-1002.
21. -1 (Ω) of the wave equation with Dirichlet boundary control acting on a portion of the boundary ∂Ω, and related problems, Appl. Math. Optim., 18 (1988), 241-277.
22. M. Yamamoto, Stability, reconstruction formula and regularization for an inverse source hyperbolic problem by a control method, Inverse Problems, 11 (1995), 481-496.
23. M. Yamamoto, On ill-posedness and a Tikhonov regularization for a multidimentional inverse hyperbolic problem, J. Math. Kyoto Univ., 36 (1996), 825-856.
24. M. Yamamoto, Uniqueness and stability in multidimentional hyperbolic inverse problems, J. Math. Pures Appl., 78 (1999), 65-98.
25. J. Yan, Contrôlabilité exacte pour des systèmes à mémoire, Rev. Mat. Univ. Complut. Madrid, 5 (1992), 319-327.
26. X. Zhang. Explicit observability inequalities for the wave equation with lower order terms by means of Carleman inequalities, SIAM J. Control Optim., 39 (2000), 812-834.