Determining the time-dependent matrix potential in a wave equation from partial boundary data

References

• Avdonin SA, Belishev MI, Ivanov SA. Boundary control and an inverse matrix problem for the equation utt−uxx+V(x)u=0.. (Russian) Mat Sb. 1991;182(3):307–331. Translation in Math. USSR-Sb. 1992;72(2):287–310.
   Google Scholar
• Khanfer A, Bukhgeĭm A. Inverse problem for one-dimensional wave equation with matrix potential. J Inverse Ill-Posed Probl. 2019;27(2):217–223. doi: https://doi.org/10.1515/jiip-2018-0053
   Google Scholar
• Blagoveshchenskii AS. On a nonselfadjoint inverse boundary-value problem in matrix form for a hyperbolic differential equation, Problemy Mat. Fiz., vyp. 5, Izdat. Leningrad. Gos. Univ., Leningrad; 1971. p. 38–61; English transl. in Topics in Math. Phys., no. 5, New York: Plenum Press; 1972.
   Google Scholar
• Eskin G, Ralston J. Inverse scattering problems for the Schrödinger operators with external Yang–Mills potentials. Partial differential equations and their applications (Toronto, ON, 1995). Providence, RI: Amer. Math. Soc.; 1997. p. 91–106. (CRM Proc. Lecture Notes; 12).
   Google Scholar
• Eskin G, Ralston J. Inverse boundary value problems for systems of partial differential equations. (English summary) Recent development in theories & numerics. River Edge, NJ: World Sci. Publ.; (2003). p. 105–113. 35R30 (35J10).
   Google Scholar
• Bukhgeĭm AL, Klibanov MV. Uniqueness in the large of a class of multidimensional inverse problems. Dokl Akad Nauk SSSR. 1981;260(2):269–272.
   Google Scholar
• Rakesh, Symes WW. Uniqueness for an inverse problem for the wave equation. Comm Partial Differ Eq. 1988;13(1):87–96. doi: https://doi.org/10.1080/03605308808820539
   Web of Science ®Google Scholar
• Sylvester J, Uhlmann G. A global uniqueness theorem for an inverse boundary value problem. Ann Math (2). 1987;125(1):153–169. doi: https://doi.org/10.2307/1971291
   Web of Science ®Google Scholar
• Ramm AG, Sjöstrand J. An inverse problem of the wave equation. Math Z. 1991;206(1):119–130. doi: https://doi.org/10.1007/BF02571330
   Web of Science ®Google Scholar
• Isakov V. Completeness of products of solutions and some inverse problems for PDE. J Differ Eq. 1991;92(2):305–316. doi: https://doi.org/10.1016/0022-0396(91)90051-A
   Google Scholar
• Kian Y. Unique determination of a time-dependent potential for wave equations from partial data. Ann Inst H Poincaré Anal Non Linéaire. 2017;34:973–990. doi: https://doi.org/10.1016/j.anihpc.2016.07.003
   Google Scholar
• Kian Y, Oksanen L. Recovery of time-dependent coefficient on Riemanian manifold for hyperbolic equations. Int Math Res Notices. 2019 Aug;2019(16):5087–5126. https://doi.org/https://doi.org/10.1093/imrn/rnx263.
   Web of Science ®Google Scholar
• Anikonov YE, Cheng J, Yamamoto M. A uniqueness result in an inverse hyperbolic problem with analyticity. Eur J Appl Math. 2004;15(5):533–543. doi: https://doi.org/10.1017/S0956792504005649
   Google Scholar
• Belishev MI. Boundary control method in dynamical inverse problems – an introductory course. In: Dynamical inverse problems: theory and application, New York, Vienna: SpringerWien; 2011. p. 85–150. (ISM Courses and Lect.; 529).
   Google Scholar
• Bellassoued M, Jellali D, Yamamoto M. Lipschitz stability for a hyperbolic inverse problem by finite local boundary data. Appl Anal. 2006;85(10):1219–1243. doi: https://doi.org/10.1080/00036810600787873
   Google Scholar
• Bellassoued M, Yamamoto M. Determination of a coefficient in the wave equation with a single measurement. Appl Anal. 2008;87(8):901–920. doi: https://doi.org/10.1080/00036810802369249
   Web of Science ®Google Scholar
• Bellassoued M, Jellali D, Yamamoto M. Stability estimate for the hyperbolic inverse boundary value problem by local Dirichlet-to-Neumann map. J Math Anal Appl. 2008;343(2):1036–1046. doi: https://doi.org/10.1016/j.jmaa.2008.01.098
   Web of Science ®Google Scholar
• Ben Aicha I. Stability estimate for hyperbolic inverse problem with time-dependent coefficient. Inverse Probl. 2015;31:125010.
   Google Scholar
• Katchalov A, Kurylev Y, Lassas M. Inverse boundary spectral problems. Boca Raton, FL: Chapman & Hall/CRC; 2001. (Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics; 123).
   Google Scholar
• Kian Y. Stability in the determination of a time-dependent coefficient for wave equations from partial data. J Math Anal Appl. 2016;436:408–428. doi: https://doi.org/10.1016/j.jmaa.2015.12.018
   Google Scholar
• Stefanov PD, Yang Y. The inverse problem for the Dirichlet-to-Neumann map on Lorentzian manifolds. Anal PDE. 2018;11(6):1381–1414. doi: https://doi.org/10.2140/apde.2018.11.1381
   Web of Science ®Google Scholar
• Stefanov PD. Inverse scattering problem for the wave equation with time-dependent potential. J Math Anal Appl. 1989;140(2):351–362. doi: https://doi.org/10.1016/0022-247X(89)90069-3
   Web of Science ®Google Scholar
• Lions JL, Magenes E. Non-homogeneous boundary value problems and applications. Vol. I. Paris: Dunod; 1968.
   Google Scholar
• Lions JL. Équations différentielles opérationnelles et problèmes aux limites (French) Die Grundlehren der mathematischen Wissenschaften, Bd. 111. Berlin-Göttingen-Heidelberg: Springer-Verlag; 1961. ix+292 pp.
   Google Scholar
• Lasiecka I, Lions JL, Triggiani R. Nonhomogeneous boundary value problems for second order hyperbolic operators. J Math Pures Appl (9). 1986;65(2):149–192.
   Web of Science ®Google Scholar
• Bukhgeĭm AL, Uhlmann G. Recovering a potential from partial Cauchy data. Comm Partial Differ Eq. 2002;27(34):653–668. doi: https://doi.org/10.1081/PDE-120002868
   Google Scholar
• Mourad Bellassoued and Imen Rassas. Stability estimate in the determination of a time-dependent coefficient for hyperbolic equation by partial Dirichlet-to-Neumann map. Appl Anal. 2019;98(15):2751–2782. doi: https://doi.org/10.1080/00036811.2018.1471206
   Google Scholar
• Hu G, Kian Y. Determination of singular time-dependent coefficients for wave equations from full and partial data. Inverse Probl Imaging. 2018;12:745–772. doi: https://doi.org/10.3934/ipi.2018032
   Google Scholar
• Kian Y. Recovery of time-dependent damping coefficients and potentials appearing in wave equations from partial data. SIAM J Math Anal. 2016;48(6):4021–4046. doi: https://doi.org/10.1137/16M1076708
   Web of Science ®Google Scholar
• Krishnan VP, Vashisth M. An inverse problem for the relativistic Schrödinger equation with partial boundary data. Appl Anal. 2019;1–21. doi:https://doi.org/10.1080/00036811.2018.1549321
   Google Scholar
• Cipolatti R, Yamamoto M. An inverse problem for a wave equation with arbitrary initial values and a finite time of observations. Inverse Probl. 2011;27(9):095006, 15 pp. doi: https://doi.org/10.1088/0266-5611/27/9/095006
   Web of Science ®Google Scholar
• Eskin G. A new approach to hyperbolic inverse problems. Inverse Probl. 2006;22(3):815–831. doi: https://doi.org/10.1088/0266-5611/22/3/005
   Web of Science ®Google Scholar
• Eskin G. Inverse hyperbolic problems with time-dependent coefficients. Comm Partial Differ Eq. 2007;32(10–12):1737–1758. doi: https://doi.org/10.1080/03605300701382340
   Google Scholar
• Eskin G. A new approach to hyperbolic inverse problems. II. Global step. Inverse Probl. 2007;23(6):2343–2356. doi: https://doi.org/10.1088/0266-5611/23/6/004
   Web of Science ®Google Scholar
• Hussein SO, Lesnic D, Yamamoto M. Reconstruction of space-dependent potential and/or damping coefficients in the wave equation. Comput Math Appl. 2017;74(6):1435–1454. doi: https://doi.org/10.1016/j.camwa.2017.06.030
   Web of Science ®Google Scholar
• Imanuvilov OY, Yamamoto M. Global uniqueness and stability in determining coefficients of wave equations. Comm Partial Differ Eq. 2001;26(7–8):1409–1425. doi: https://doi.org/10.1081/PDE-100106139
   Google Scholar
• Isakov V. An inverse hyperbolic problem with many boundary measurements. Comm Partial Differ Eq. 1991;16(6–7):1183–1195. doi: https://doi.org/10.1080/03605309108820794
   Web of Science ®Google Scholar
• Ramm AG, Rakesh. Property C and an inverse problem for a hyperbolic equation. J Math Anal Appl. 1991;156(1):209–219. doi: https://doi.org/10.1016/0022-247X(91)90391-C
   Web of Science ®Google Scholar
• Salazar R. Determination of time-dependent coefficients for a hyperbolic inverse problem. Inverse Probl. 2013;29(9):095015. doi: https://doi.org/10.1088/0266-5611/29/9/095015
   Web of Science ®Google Scholar