Publication Cover
Applicable Analysis
An International Journal
Volume 92, 2013 - Issue 10
202
Views
22
CrossRef citations to date
0
Altmetric
Articles

Lipschitz stability in an inverse problem for the Kuramoto–Sivashinsky equation

, , &
Pages 2084-2102 | Received 04 May 2012, Accepted 24 Jul 2012, Published online: 29 Aug 2012

References

  • Bukhgeĭm , AL and Klibanov , MV . 1981 . Uniqueness in the large of a class of multidimensional inverse problems . Dokl. Akad. Nauk SSSR , 260 ( 2 ) : 269 – 272 .
  • Beilina , L and Klibanov , M . 2012 . Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems , New York , , USA : Springer .
  • Klibanov , MV . 1992 . Inverse problems and Carleman estimates . Inverse Probl. , 8 ( 4 ) : 575 – 596 .
  • Klibanov , MV and Malinsky , J . 1991 . Newton–Kantorovich method for three-dimensional potential inverse scattering problem and stability of the hyperbolic Cauchy problem with time-dependent data . Inverse Probl. , 7 ( 4 ) : 577 – 596 .
  • Puel , J-P and Yamamoto , M . 1996 . On a global estimate in a linear inverse hyperbolic problem . Inverse Probl. , 12 ( 6 ) : 995 – 1002 .
  • Yu. Imanuvilov , O and Yamamoto , M . 1998 . Lipschitz stability in inverse parabolic problems by the Carleman estimate . Inverse Probl. , 14 ( 5 ) : 1229 – 1245 .
  • Roques , L and Cristofol , M . 2010 . On the determination of the nonlinearity from localized measurements in a reaction-diffusion equation . Nonlinearity , 23 ( 3 ) : 675 – 686 .
  • Klibanov , MV and Timonov , A . 2004 . Carleman estimates for Coefficient Inverse Problems and Numerical Applications. , Utrecht : Inverse and Ill-posed Problems Series. VSP .
  • Benabdallah , A , Gaitan , P and Le Rousseau , J . 2007 . Stability of discontinuous diffusion coefficients and initial conditions in an inverse problem for the heat equation . SIAM J. Control Optim. , 46 ( 5 ) : 1849 – 1881 .
  • Cristofol , M , Gaitan , P and Ramoul , H . 2006 . Inverse problems for a 2 × 2 reaction–diffusion system using a Carleman estimate with one observation . Inverse Probl. , 22 ( 5 ) : 1561 – 1573 .
  • L.I. Ignat, A.F. Pazoto and L. Rosier, Inverse Problem for the heat equation and the Schrodinger equation on a tree, Inverse Probl. 28(1) (2012), 015011, 30 pp
  • Boulakia , M , Grandmont , C and Osses , A . 2009 . Some inverse stability results for the bistable reaction–diffusion equation using Carleman inequalities . C. R. Math. Acad. Sci. Paris , 347 ( 11–12 ) : 619 – 622 .
  • Egger , H , Engl , HW and Klibanov , MV . 2005 . Global uniqueness and Hölder stability for recovering a nonlinear source term in a parabolic equation . Inverse Probl. , 21 ( 1 ) : 271 – 290 .
  • Bellassoued , M and Yamamoto , M . 2006 . Logarithmic stability in determination of a coefficient in an acoustic equation by arbitrary boundary observation . J. Math. Pures Appl. (9) , 85 ( 2 ) : 193 – 224 .
  • Dos Santos Ferreira , D , Kenig , CE , Salo , M and Uhlmann , G . 2009 . Limiting Carleman weights and anisotropic inverse problems . Invent. Math. , 178 ( 1 ) : 119 – 171 .
  • Baudouin , L and Puel , J-P . 2002 . Uniqueness and stability in an inverse problem for the Schrödinger equation . Inverse Probl. , 18 ( 6 ) : 1537 – 1554 .
  • Kuramoto , Y and Tsuzuki , T . 1975 . On the formation of dissipative structures in reaction–diffusion systems . Prog. Theor. Phys , 54 : 687 – 99 .
  • Sivashinsky , GI . 1977 . Nonlinear analysis of hydrodynamic instability in laminar flames I: Derivation of basic equations . Acta Astronaut. , 4 : 1177 – 1206 .
  • Liu , W-J and Krstić , M . 2001 . Stability enhancement by boundary control in the Kuramoto–Sivashinsky equation . Nonlinear Anal. Ser. A: Theory Methods , 43 : 485 – 507 .
  • Cerpa , E and Mercado , A . 2011 . Local exact controllability to the trajectories of the 1-D Kuramoto–Sivashinsky equation . J. Differ. Eqns. , 250 ( 4 ) : 2024 – 2044 .
  • Hu , C and Temam , R . 2001 . Robust control of the Kuramoto–Sivashinsky equation . Dyn. Contin. Discr. Impuls. Syst. Ser. B Appl. Algorithms , 8 : 315 – 338 .
  • Armaou , A and Christofides , PD . 2000 . Feedback control of the Kuramoto–Sivashinsky equation . Physica. D , 137 : 49 – 61 .
  • Christofides , PD and Armaou , A . 2000 . Global stabilization of the Kuramoto–Sivashinsky equation via distributed output feedback control . Syst. Control Lett. , 39 : 283 – 294 .
  • Cerpa , E . 2010 . Null controllability and stabilization of a linear Kuramoto–Sivashinsky equation . Commun. Pure Appl. Anal. , 9 : 91 – 102 .
  • Coron , J-M and Guerrero , S . 2005 . Singular optimal control: A linear 1-D parabolic-hyperbolic example . Asymptotic Anal. , 44 : 237 – 257 .
  • Fernández-Cara , E and Guerrero , S . 2006 . Global Carleman inequalities for parabolic systems and applications to controllability . SIAM J. Control Optim. , 45 ( 4 ) : 1395 – 1446 .
  • Cazenave , T and Haraux , A . 1998 . An Introduction to Semilinear Evolution Equations , New York , , USA : Oxford University Press .
  • Renardy , M and Rogers , R . 2004 . “ An Introduction to Partial Differential Equations ” . In Texts in Applied Mathematics , New York : Springer-Verlag .

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.