Publication Cover
Applicable Analysis
An International Journal
Volume 94, 2015 - Issue 11
120
Views
4
CrossRef citations to date
0
Altmetric
Articles

Inverse problem on a tree-shaped network: unified approach for uniqueness

&
Pages 2370-2395 | Received 21 Jul 2014, Accepted 04 Nov 2014, Published online: 02 Dec 2014

References

  • Dáger R, Zuazua E. Wave propagation, observation and control in 1-d flexible multi-structures. Vol. 50, Mathématiques & applications (Berlin). Berlin: Springer-Verlag; 2006.
  • Alexander S. Superconductivity of networks. A percolation approach to the effects of disorder. Phys. Rev. B. 1983;23:1541–1557.
  • Exner P, Post O. A general approximation of quantum graph vertex couplings by scaled Schrödinger operators on thin branched manifolds. Commun. Math. Phys. 2013;322:207–227.
  • Kuchment P. Graph models for waves in thin structures. Waves Rand. Media. 2002;12:R1–R24.
  • Ignat LI, Pazoto AF, Rosier L. Inverse problem for the heat equation and the Schrödinger equation on a tree. Inverse Probl. 2012;28:015011.
  • Avdonin S, Leugering G, Mikhaylov V. On an inverse problem for tree-like networks of elastic strings. ZAMM Z. Angew. Math. Mech. 2010;90:136–150.
  • Belishev MI. Boundary spectral inverse problem on a class of graphs (trees) by the boundary control method. Inverse Probl. 2004;20:647–672.
  • Baudouin L, Crépeau E, Valein J. Global Carleman estimate on a network for the wave equation and application to an inverse problem. Math. Control Relat. Fields. 2011;1:307–330.
  • Lions J-L, Magenes E. Non-homogeneous boundary value problems and applications. New York (NY): Springer-Verlag; 1972; Translated from the French by P. Kenneth. Die Grundlehren der mathematischen Wissenschaften. Vol. 182.
  • Lagnese J, Leugering G, Schmidt EJPG. Modeling, analysis and control of dynamic elastic multi-link structures. Systems & control: foundations & applications. Boston (MA): Birkhäuser Boston Inc.; 1994.
  • Bukhgeĭm AL, Klibanov MV. Uniqueness in the large of a class of multidimensional inverse problems. Dokl. Akad. Nauk SSSR. 1981;260:269–272.
  • Klibanov MV. Inverse problems and Carleman estimates. Inverse Probl. 1992;8:575–596.
  • Yamamoto M. Uniqueness and stability in multidimensional hyperbolic inverse problems. J. Math. Pures Appl. 1999;78:65–98.
  • Klibanov MV. Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems. J. Inverse Ill-Posed Probl. 2013;21:477–560.
  • Isakov V. Inverse problems for partial differential equations. 2nd ed. Vol. 127, Applied mathematical sciences. New York (NY): Springer; 2006.
  • Baudouin L, Mercado A, Osses A. A global Carleman estimate in a transmission wave equation and application to a one-measurement inverse problem. Inverse Probl. 2007;23:257–278.
  • Imanuvilov OY, Yamamoto M. Global Lipschitz stability in an inverse hyperbolic problem by interior observations. Inverse Probl. 2001;17:717–728. Special issue to celebrate Pierre Sabatier’s 65th birthday (Montpellier, 2000).
  • Imanuvilov OY, Yamamoto M. Global uniqueness and stability in determining coefficients of wave equations. Comm. Par. Differ. Equ. 2001;26:1409–1425.
  • Puel J-P, Yamamoto M. Generic well-posedness in a multidimensional hyperbolic inverse problem. J. Inverse Ill-Posed Probl. 1997;5:55–83.
  • Yamamoto M, Zhang X. Global uniqueness and stability for a class of multidimensional inverse hyperbolic problems with two unknowns. Appl. Math. Optim. 2003;48:211–228.
  • Zuazua E. Control and stabilization of waves on 1-d networks. Modelling and optimisation of flows on networks of lecture notes in mathematics, CIME foundation subseries. Heidelberg: Springer; 2013. p. 463–493.
  • Dáger R. Observation and control of vibrations in tree-shaped networks of strings. SIAM J. Control Optim. 2004;43:590–623. electronic.
  • Dáger R, Zuazua E. Controllability of star-shaped networks of strings. Mathematical and numerical aspects of wave propagation (Santiago de Compostela, 2000). Philadelphia (PA): SIAM; 2000. p. 1006–1010.
  • Schmidt EJPG. On the modelling and exact controllability of networks of vibrating strings. SIAM J. Control Optim. 1992;30:229–245.
  • Lions J-L. Contrôlabilité exacte, stabilisation et perturbations de systèmes distribués [Exact controllability, perturbations and stabilization of distributed systems]. Tome 1. Contrôlabilité exacte. Vol. 1. Paris: Masson; 1988.
  • Klibanov MV, Timonov A. Carleman estimates for coefficient inverse problems and numerical applications. Inverse and ill-posed problems series. Utrecht: VSP; 2004.
  • Lavrent’ev MM, Romanov VG, Shishatskiĭ SP. Ill-posed problems of mathematical physics and analysis. Translations of mathematical monographs. Providence (RI): American Mathematical Society; 1986.
  • Romanov VG. Inverse problems of mathematical physics. Utrecht: VNU Science Press, b.v.; 1987.
  • Pazy A. Semigroups of linear operators and applications to partial differential equations. Vol. 44, Applied mathematical sciences. New York (NY): Springer-Verlag; 1983.
  • Tanabe H. Equations of evolution. Vol. 6, Monographs and studies in mathematics. Boston (MA): Pitman (Advanced Publishing Program); 1979.
  • Yuan G, Yamamoto M. Lipschitz stability in inverse problems for a Kirchhoff plate equation. Asymptot. Anal. 2007;53:29–60.
  • Yuan G, Yamamoto M. Carleman estimates for the Schrödinger equation and applications to an inverse problem and an observability inequality. Chin. Ann. Math. Ser. B. 2010;31:555–578.
  • Baudouin L, Puel J-P. Uniqueness and stability in an inverse problem for the Schrödinger equation. Inverse Probl. 2002;18:1537–1554.
  • Mercado A, Osses A, Rosier L. Inverse problems for the Schrödinger equation via Carleman inequalities with degenerate weights. Inverse Probl. 2008;24:015017.
  • Baudouin L, De Buhan M, Ervedoza S. Global Carleman estimates for waves and applications. Comm. Par. Differ. Equ. 2013;38:823–859.
  • Hörmander L. Linear partial differential operator. Vol. 116, Die Grundlehren der mathematischen Wissenschaften. Berlin: Springer; 1963.
  • Bellassoued M, Yamamoto M. Carleman estimates and applications to inverse problems for hyperbolic systems. Springer monographs in mathematics. Berlin: Springer; 2014.

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.