On Carleman estimates for hyperbolic equations

Asymptotic Analysis

Volumn 32, Issue 3-4, 2002, Pages 185-220

  Imanuvilov, Oleg Yu ^a  

^a IOWA STATE UNIVERSITY   (United States)

Author keywords

Carleman estimate; Hyperbolic equation

Indexed keywords

EID: 0036975369     PISSN: 09217134     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (33)

1. G. Avalos and I. Lasiecka, Exact-approximate boundary reachability of thermoelastic plates under variable thermal coupling, Inverse Problems 16(4) (2000), 978-996.
2. C. Bardos, J. 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.
3. C. Bardos, L. Halperin G. Lebeau, J. Rauch and E. Zuazua, Stabilization de l'equation des ondes au moyen d'un feedback portant sur la condition aux limites de Dirichlet, Asymptotic Anal. 4 (1991), 285-291.
4. T. Carleman, Sur un probleme d'unicité pour les systèmes d'èquations aux derivées partielles á deux variables independantes, Ark. Mat. Astr. Fys. B 26 (1939), 1-9.
5. D. Chae, O. Imanuvilov and S.M. Kim, Exact controllability for the semilinear parabolic equations with Neumann boundary conditions, J. Dynamical Control Systems 2 (1996), 449-483.
6. C. Fabre, Résultats d'unicité pour les équations de Stokes et applications au contrôle, C. R. Acad. Sci. Paris Sér. I 322 (1996), 1191-1196.
7. C. Fabre and G. Lebeau, Prolongement unique des solutions de l'équation de Stokes, Comm. Partial Differential Equations 21(3-4) (1996), 573-596.
8. L. Hörmander, Linear Partial Differential Operators, Springer, New York, 1969.
9. L. Hörmander, The Analysis of Linear Partial Differential Operators III, Springer, New York, 1983.
10. O. Imanuvilov, Some problems of optimization and exact controllability, Thesis, Moscow, 1991 (in Russian).
11. O. Imanuvilov and M. Yamamoto, On Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations. Preprint UTMS 98-46, 1998.
12. O. Imanuvilov and M. Yamamoto, Carleman estimate for parabolic equations in a Sobolev spaces of negative order and its applications, Control Nonlinear Distributed Parameter Systems 218 (2001), 113-137.
13. V. Isakov, Carleman estimates in an anisotropic case and applications, J. Differential Equations 105 (1993), 217-238.
14. V. Isakov, On uniqueness in the lateral Cauchy problem with multiple characteristics, J. Differential Equations 134 (1997), 134-147.
15. V. Isakov and M. Yamamoto, Lipschitz stability in a lateral Cauchy problem of a hyperbolic equation with the homogeneous Neumann boundary condition. Preprint, UTMS 99-36, Graduate School of Mathematical Society, The University of Tokyo, 1999.
16. V. Isakov and M. Yamamoto, Carleman estimate with the Neumann boundary conditions and its applications to the observability and inverse hyperbolic problems, in: Contemp. Math., Vol. 268, 2000, pp. 191-225.
17. O.A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics, Springer, 1985.
18. I. Lasiecka and C. Lebiedzik, Uniform stability in structural acoustic systems with thermal effects and nonlinear boundary damping, Control Cybernet. 29(3) (1999), 557-581.
19. I. Lasiecka and R. Triggiani, Controllability of semilinear abstract systems with application to waves and plates boundary control problems, Appl. Math. Optim. 23 (1991), 109-154.
20. I. Lasiecka and R. Triggiani, Optimal regularity, exact controllability and uniform stabilization of Schrödinger equations with Dirichlet control, Differential Integral Equations 5 (1992), 521-535.
21. I. Lasiecka and R. Triggiani, Carleman estimates and exact boundary controllability for a system of coupled, nonconservative second order hyperbolic equations, J. Math. Anal. Appl. 225 (1999), 13-57.
22. I. Lasiecka, R. Triggiani and P. Yao, Inverse/observability estimates for second-order hyperbolic equations with variable coefficients, Appl. Math. Optim. 10 (1983), 275-286.
23. -1(Ω) for the second-order hyperbolic equations with variable coefficients, in: Control of Distributed Parameter and Stochastic Systems, E.S. Chen, X. Li, J. Yong and X. Zhou, eds, 1999, pp. 71-79.
24. I. Lasiecka, R. Triggiani and X. Zhang, Nonconservative wave equations with unobserved Neumann B.C.: Global uniqueness and observability in one short, in: Contemp. Math., Vol. 268 (2000), pp. 227-325.
25. M. Lavrent'ev, V. Romanov and S. Shishatskii, Ill-Posed Problems of Mathematical Physics and Analysis, Amer. Math. Soc., Providence, RI, 1986.
26. J.-L. Lions, Contrôlabilite' Exacte, Perturbation et Stabilization de Systemes Distribués, Vol. 1, Masson, Paris, 1988.
27. A. Ruiz, Unique continuation for weak solutions of the wave equation plus a potential, J. Math. Pures Appl. 71(6) (1992), 455-467.
28. J.-C. Saut and B. Scheurer, Unique continuation for some evolution equations, J. Differential Equations 66 (1987), 118-139.
29. C.D. Sogge, Unique continuation theorem for second order parabolic differential operators, Ark. Mat. 28 (1990), 159-182.
30. C.D. Sogge, Oscillatory integrals, Carleman inequalities and unique continuation for second order elliptic operators, J. Amer. Math. Soc. 2 (1989), 491-516.
31. D. Tataru, Carleman estimates and unique continuation for solutions to boundary value problems, J. Math. Pures Appl. 76 (1996), 367-408.
32. θ spaces and unique continuation for solutions to the semilinear wave equation, Comm. Partial Differential Equations 21(5-6) (1996), 841-887.
33. D. Tataru, Boundary controllability for conservative P.D.E., Appl. Math. Optim. 31 (1995), 257-295.