Free access
Issue
ESAIM: COCV
Volume 15, Number 3, July-September 2009
Page(s) 525 - 554
DOI http://dx.doi.org/10.1051/cocv:2008043
Published online 19 July 2008
  1. R.A. Adams, Sobolev Spaces. Academic Press, New York (1975).
  2. K.A. Ames and B. Straughan, Non-standard and Improperly Posed Problems. Academic Press, San Diego (1997).
  3. L. Baudouin and J.-P. Puel, Uniqueness and stability in an inverse problem for the Schrödinger equation. Inverse Probl. 18 (2002) 1537–1554. [CrossRef] [MathSciNet] [PubMed]
  4. M. Bellassoued, Global logarithmic stability in inverse hyperbolic problem by arbitrary boundary observation. Inverse Probl. 20 (2004) 1033–1052. [CrossRef]
  5. M. Bellassoued and M. Yamamoto, Logarithmic stability in determination of a coefficient in an acoustic equation by arbitrary boundary observation. J. Math. Pures Appl. 85 (2006) 193–224. [CrossRef] [MathSciNet]
  6. H. Brezis, Analyse Fonctionnelle. Masson, Paris (1983).
  7. A.L. Bukhgeim, Introduction to the Theory of Inverse Probl. VSP, Utrecht (2000).
  8. A.L. Bukhgeim and M.V. Klibanov, Global uniqueness of a class of multidimensional inverse problems. Soviet Math. Dokl. 24 (1981) 244–247.
  9. D. Chae, O.Yu. Imanuvilov and S.M. Kim, Exact controllability for semilinear parabolic equations with Neumann boundary conditions. J. Dyn. Contr. Syst. 2 (1996) 449–483. [CrossRef] [MathSciNet]
  10. J. Cheng and M. Yamamoto, One new strategy for a priori choice of regularizing parameters in Tikhonov's regularization. Inverse Probl. 16 (2000) L31–L38. [CrossRef]
  11. P.G. Danilaev, Coefficient Inverse Problems for Parabolic Type Equations and Their Application. VSP, Utrecht (2001).
  12. A. Elayyan and V. Isakov, On uniqueness of recovery of the discontinuous conductivity coefficient of a parabolic equation. SIAM J. Math. Anal. 28 (1997) 49–59. [CrossRef] [MathSciNet]
  13. M.M. Eller and V. Isakov, Carleman estimates with two large parameters and applications. Contemp. Math. 268 (2000) 117–136.
  14. C. Fabre, J.-P. Puel and E. Zuazua, Approximate controllability of the semilinear heat equation. Proc. Royal Soc. Edinburgh 125A (1995) 31–61.
  15. A.V. Fursikov and O.Yu. Imanuvilov, Controllability of Evolution Equations, in Lecture Notes Series 34, Seoul National University, Seoul, South Korea (1996).
  16. D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin (2001).
  17. R. Glowinski and J.L. Lions, Exact and approximate controllability for distributed parameter systems. Acta Numer. 3 (1994) 269–378. [CrossRef]
  18. L. Hörmander, Linear Partial Differential Operators. Springer-Verlag, Berlin (1963).
  19. O.Yu. Imanuvilov, Controllability of parabolic equations. Sb. Math. 186 (1995) 879–900. [CrossRef] [MathSciNet]
  20. O.Yu. Imanuvilov and M. Yamamoto, Lipschitz stability in inverse parabolic problems by the Carleman estimate. Inverse Probl. 14 (1998) 1229–1245. [CrossRef] [MathSciNet]
  21. O.Yu. Imanuvilov and M. Yamamoto, Global Lipschitz stability in an inverse hyperbolic problem by interior observations. Inverse Probl. 17 (2001) 717–728. [CrossRef] [MathSciNet]
  22. O.Yu. Imanuvilov and M. Yamamoto, Carleman estimate for a parabolic equation in a Sobolev space of negative order and its applications, in Control of Nonlinear Distributed Parameter Systems, Marcel Dekker, New York (2001) 113–137.
  23. O.Yu. Imanuvilov and M. Yamamoto, Determination of a coefficient in an acoustic equation with a single measurement. Inverse Probl. 19 (2003) 151–171.
  24. O.Yu. Imanuvilov and M. Yamamoto, Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations. Publ. RIMS Kyoto Univ. 39 (2003) 227–274. [CrossRef] [MathSciNet]
  25. V. Isakov, Inverse Problems for Partial Differential Equations. Springer-Verlag, Berlin (1998), (2005).
  26. V. Isakov and S. Kindermann, Identification of the diffusion coefficient in a one-dimensional parabolic equation. Inverse Probl. 16 (2000) 665–680. [CrossRef]
  27. M. Ivanchov, Inverse Problems for Equations of Parabolic Type. VNTL Publishers, Lviv, Ukraine (2003).
  28. A. Khaĭdarov, Carleman estimates and inverse problems for second order hyperbolic equations. Math. USSR Sbornik 58 (1987) 267–277. [CrossRef]
  29. M.V. Klibanov, Inverse problems in the “large” and Carleman bounds. Diff. Equ. 20 (1984) 755–760.
  30. M.V. Klibanov, Inverse problems and Carleman estimates. Inverse Probl. 8 (1992) 575–596. [CrossRef] [MathSciNet]
  31. M.V. Klibanov, Estimates of initial conditions of parabolic equations and inequalities via lateral Cauchy data. Inverse Probl. 22 (2006) 495–514. [CrossRef]
  32. M.V. Klibanov and A.A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications. VSP, Utrecht (2004).
  33. M.V. Klibanov and M. Yamamoto, Lipschitz stability of an inverse problem for an accoustic equation. Appl. Anal. 85 (2006) 515–538. [CrossRef] [MathSciNet]
  34. M.M. Lavrent'ev, V.G. Romanov and ShishatFormula skiĭ, Ill-posed Problems of Mathematical Physics and Analysis. American Mathematical Society Providence, Rhode Island (1986).
  35. J.L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Springer-Verlag, Berlin (1972).
  36. L.E. Payne, Improperly Posed Problems in Partial Differential Equations. SIAM, Philadelphia (1975).
  37. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York (1983).
  38. J.C. Saut and B. Scheurer, Unique continuation for some evolution equations. J. Diff. Eq. 66 (1987) 118–139. [CrossRef] [MathSciNet]
  39. E.J.P.G. Schmidt and N. Weck, On the boundary behavior of solutions to elliptic and parabolic equations – with applications to boundary control for parabolic equations. SIAM J. Contr. Opt. 16 (1978) 593–598. [CrossRef]
  40. M. Yamamoto, Uniqueness and stability in multidimensional hyperbolic inverse problems. J. Math. Pures Appl. 78 (1999) 65–98. [CrossRef] [MathSciNet]
  41. M. Yamamoto and J. Zou, Simultaneous reconstruction of the initial temperature and heat radiative coefficient. Inverse Probl. 17 (2001) 1181–1202. [CrossRef]