Recovering damping and potential coefficients for an inverse nonhomogeneous secondorder hyperbolic problem via a localized Neumann boundary trace
Pages: 5217  5252,
Issue 11/12,
November/December
2013
doi:10.3934/dcds.2013.33.5217 Abstract
References
Full text (601.1K)
Related Articles
Shitao Liu  Department of Mathematics and Statistics, University of Helsinki, FI00014 Helsinki, Finland (email)
Roberto Triggiani  Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, United States (email)
1 
A. Bukhgeim, J. Cheng, V. Isakov and M. Yamamoto, Uniqueness in determining damping coefficients in hyperbolic equations, Analytic Extension Formulas and their Applications, Kluwer, Dordrecht (2001), 2746. 

2 
A. Bukhgeim and M. Klibanov, Global uniqueness of a class of multidimensional inverse problem, Sov., Math.Dokl., 24 (1981), 244247. 

3 
T. Carleman, Sur un problème d'unicité pour les systèmes d'équations aux derivées partielles à deux variables independantes, Ark. Mat. Astr. Fys., 2B (1939), 19. 

4 
L.F.Ho, Observabilite frontiere de l'equation des ondes, Comptes Rendus de l'Academie des Sciences de Paris, 302 (1986), 443446. 

5 
O. Imanuvilov and M. Yamamoto, Global Lipschitz stability in an inverse hyperbolic problem by interior observations, Inverse Problems, 17 (2001), 717728. 

6 
O. Imanuvilov and M. Yamamoto, Global uniqueness and stability in determining coefficients of wave equations, Comm. Partial Differential Equations, 26 (2001), 14091425. 

7 
V. Isakov, "Inverse Problems for Partial Differential Equations," First Edition, Springer, New York, 1998. 

8 
V. Isakov, "Inverse Problems for Partial Differential Equations," Second Edition, Springer, New York, 2006. 

9 
V. Isakov, "Inverse Source Problems," American Mathematical Society, 2000. 

10 
V. Isakov and M. Yamamoto, Carleman estimate with the Neumann boundary condition and its application to the observability inequality and inverse hyperbolic problems, Contemp. Math., 268 (2000), 191225. 

11 
V. Isakov and M. Yamamoto, Stability in a wave source problem by Dirichlet data on subboundary, J. of Inverse & IllPosed Problems, 11 (2003), 399409. 

12 
M. Klibanov, Inverse problems and Carleman estimates, Inverse Problems, 8 (1992), 575596. 

13 
M. Klibanov and A. Timonov, "Carleman Estimates For Coefficient Inverse Problems and Numerical Applications," VSP, Utrecht, 2004. 

14 
I. Lasiecka, J. L. Lions and R. Triggiani, Non homogeneous boundary value problems for secondorder hyperbolic operators, J. Math. Pures Appl., 65 (1986), 149192. 

15 
I. Lasiecka and R. Triggiani, A cosine operator approach to modeling $L_2(0,T;L_2(\Omega))$ boundary input hyperbolic equations, Appl. Math. & Optimiz., 7 (1981), 3583. 

16 
I. Lasiecka and R. Triggiani, Regularity of hyperbolic equations under $L_2(0,T;L_2(\Gamma))$Dirichlet boundary terms, Appl. Math. & Optimiz., 10 (1983), 275286. 

17 
I. Lasiecka and R. Triggiani, Exact controllability of the wave equation with Neumann boundary control, Appl. Math. & Optimiz., 19 (1989), 243290. 

18 
I. Lasiecka and R. Triggiani, Recent advances in regularity of secondorder hyperbolic mixed problems and applications, Dynamics Reported, SpringerVerlag, 3 (1994), 104158. 

19 
I. Lasiecka and R. Triggiani, Carleman estimates and exact controllability for a system of coupled, nonconservative secondorder hyperbolic equations, Marcel Dekker Lectures Notes Pure Appl. Math., 188 (1997), 215245. 

20 
I. Lasiecka and R. Triggiani, "Control Theory for Partial Differential Equations: Continuous and Approximation Theories," 2, Encyclopedia of Mathematics and its Applications Series, Cambridge University Press, 2000. 

21 
I. Lasiecka, R. Triggiani and P. F. Yao, Inverse/observability estimates for secondorder hyperbolic equations with variable coefficients, J. Math. Anal. Appl., 235 (1999), 1357. 

22 
I. Lasiecka, R. Triggiani and P. F. Yao, An observability estimate in $L_2(\Omega)\times H^{1}(\Omega)$ for second order hyperbolic equations with variable coefficients, Control of Distributed Parameter and Stochastic Systems, Kluwer (1999), 7179, (eds. S. Chen, X. Li, J. Yong and X. Zhou). 

23 
I. Lasiecka, R. Triggiani and X. Zhang, Nonconservative wave equations with unobserved Neumann B.C.: Global uniqueness and observability in one shot, Contemp. Math., 268 (2000), 227325. 

24 
M. M. Lavrentev, V. G. Romanov and S. P. Shishataskii, "IllPosed Problems of Mathematical Physics and Analysis," Amer. Math. Soc., Providence, RI, 64 (1986). 

25 
J. L. Lions, "Controlabilite Exacte," Perturbations et Stabilisation de Systemes Distribues, 1, Masson, Paris, 1988. 

26 
J. L. Lions and E. Magenes, "Nonhomogeneous Boundary Value Problems and Applications," I, SpringerVerlag, Berlin, 1972. 

27 
W. Littman, "Near Optimal Time Boundary Controllability for a Class of Hyperbolic Equations," Lecture Notes in Control and Inform. Sci. 97, SpringerVerlag, Berlin, 1987, 307312, 

28 
S. Liu, Inverse problem for a structural acoustic interaction, Nonlinear Anal., 74 (2011), 26472662. 

29 
S. Liu and R. Triggiani, Global uniqueness and stability in determining the damping and potential coefficients of an inverse hyperbolic problem, Nonlinear Anal. Real World Appl., 12 (2011), 15621590. 

30 
S. Liu and R. Triggiani, Global uniqueness and stability in determining the damping coefficient of an inverse hyperbolic problem with nonhomogeneous Neumann B.C. through an additional Dirichlet boundary trace, SIAM J. Math. Anal., 43 (2011), 16311666. 

31 
S. Liu and R. Triggiani, Global uniqueness in determining electric potentials for a system of strongly coupled Schrödinger equations with magnetic potential terms, J. Inverse IllPosed Probl., 19 (2011), 223254. 

32 
S. Liu and R. Triggiani, Recovering the damping coefficients for a system of coupled wave equations with Neumann BC: Uniqueness and stability, Chin. Ann. Math. Ser B, 32 (2011), 669698. 

33 
S. Liu and R. Triggiani, Determining damping and potential coefficients of an inverse problem for a system of two coupled hyperbolic equations. Part I: Global uniqueness, Dynamical Systems and Differential Equations, DCDS Supplement (2011), Proceedings of the 8th AIMS International Conference (Dresden, Germany), 10011014. 

34 
S. Liu and R. Triggiani, Global uniqueness and stability in determining the damping coefficient of an inverse hyperbolic problem with nonhomogeneous Dirichlet B.C. through an additional localized Neumann boundary trace, Applicable Analysis, 91 (2012), Special Issue on Direct and Inverse Problems, 15511581. 

35 
S. Liu and R. Triggiani, "Boundary Control and Boundary Inverse Theory for NonHomogeneous SecondOrder Hyperbolic Equations: A Common Carleman Estimates Approach," Special Volume in Book Series of American Institute of Mathematical Sciences, under Sissa's auspices, 110 pp., to appear. 

36 
V. G. Mazya and T. O. Shaposhnikova, "Theory of Multipliers in Spaces of Differentiable Functions," Monographs and Studies in Mathematics, 23, Pitman, 1985. 

37 
D. Tataru, Apriori estimates of Carleman's type in domains with boundary, J. Math. Pures. et Appl., 73 (1994), 355387. 

38 
D. Tataru, Boundary controllability for conservative PDE's, Appl. Math. & Optimiz., 31 (1995), 257295. Based on a Ph.D. dissertation, University of Virginia, (1992). 

39 
D. Tataru, Carleman estimates and unique continuation for solutions to boundary value problems, J. Math. Pures Appl., 75 (1996), 367408. 

40 
R. Triggiani, Exact boundary controllability of $L_2(\Omega) \times H^{1}(\Omega)$ of the wave equation with Dirichlet boundary control acting on a portion of the boundary and related problems, Appl. Math. & Optimiz., 18 (1988), 241277. 

41 
R. Triggiani and P. F. Yao, Carleman estimates with no lower order terms for general Riemannian wave equations: Global uniqueness and observability in one shot, Appl. Math. & Optimiz., 46 (2002), 331375. 

42 
M. Yamamoto, Uniqueness and stability in multidimensional hyperbolic inverse problems, J. Math. Pures Appl., 78 (1999), 6598. 

Go to top
