In this article we consider the inverse problem of determining the diffusion coefficient of the heat operator in an unbounded guide using a finite number of localized observations. For this problem, we prove a stability estimate in any finite portion of the guide using an adapted Carleman inequality. The measurements are located on the boundary of a larger finite portion of the guide. A special care is required to avoid measurements on the cross-section boundaries which are inside the actual guide. This stability estimate uses a technical positivity assumption. Using arguments from control theory, we manage to remove this assumption for the inverse problem with a given non homogeneous boundary condition.
Citation: |
[1] |
A. Benabdallah, M. Cristofol, P. Gaitan and M. Yamamoto, Inverse problem for a parabolic system with two components by measurements of one component, Applicable Analysis, 88 (2009), 683-709.
doi: 10.1080/00036810802555490.![]() ![]() ![]() |
[2] |
M. Bellassoued, O. Imanuvilov and M. Yamamoto, Inverse problem of determining the density and two Lamé coefficients by boundary data, SIAM J. Math. Anal., 40 (2008), 238-265.
doi: 10.1137/070679971.![]() ![]() ![]() |
[3] |
A. L. Bukhgeim and M. V. Klibanov, Uniqueness in the Large of a Class of Multidimensional Inverse Problems, Dokl. Akad. Nauk SSSR, 260 (1981), 269-272.
![]() ![]() |
[4] |
L. Cardoulis and M. Cristofol, An inverse problem for the heat equation in an unbounded guide, Appl. Math. Lett., 62 (2016), 63-68.
doi: 10.1016/j.aml.2016.06.015.![]() ![]() ![]() |
[5] |
M. Choulli, E. M. Ouhabaz and M. Yamamoto, Stable determination of a semilinear term in a parabolic equation, Commun. Pure Appl. Anal., 5 (2006), 447-462.
doi: 10.3934/cpaa.2006.5.447.![]() ![]() ![]() |
[6] |
M. Cristofol, J. Garnier, F. Hamel and L. Roques, Uniqueness from pointwise observations in a multi-parameter inverse problem, Commun. Pure Appl. Anal., 11 (2012), 173-188.
doi: 10.3934/cpaa.2012.11.173.![]() ![]() ![]() |
[7] |
M. Cristofol, I. Kaddouri, G. Nadin and L. Roques, Coefficient determination via asymptotic spreading speeds, Inverse Problems, 30 (2014), 035005, 16pp.
doi: 10.1088/0266-5611/30/3/035005.![]() ![]() ![]() |
[8] |
M. Cristofol, S. Li and E. Soccorsi, Determining the waveguide conductivity in a hyperbolic equation from a single measurement on the lateral boundary, Math. Control and Relat. Fields, 6 (2016), 407-427.
doi: 10.3934/mcrf.2016009.![]() ![]() ![]() |
[9] |
M. Cristofol and L. Roques, Biological invasions: Deriving the regions at risk from partial measurements, Math. Biosci., 215 (2008), 158-166.
doi: 10.1016/j.mbs.2008.07.004.![]() ![]() ![]() |
[10] |
M. Cristofol and L. Roques, Stable estimation of two coefficients in a nonlinear Fisher-KPP equation, Inverse Problems, 29 (2013), 095007, 18pp.
doi: 10.1088/0266-5611/29/9/095007.![]() ![]() ![]() |
[11] |
M. Cristofol and L. Roques, Simultaneous determination of the drift and diffusion coefficients in stochastic differential equations, Inverse Problems, 33 (2017), 095006, 12pp.
doi: 10.1088/1361-6420/aa7a1c.![]() ![]() ![]() |
[12] |
M. V. de Hoop, L. Qiu and O. Scherzer, Local analysis of inverse problems: Hölder stability and iterative reconstruction, Inverse Problems, 28 (2012), 045001, 16pp.
doi: 10.1088/0266-5611/28/4/045001.![]() ![]() ![]() |
[13] |
P. DuChateau and W. Rundell, Unicity in an inverse problem for an unknown reaction term in a reaction-diffusion equation, J. Differential Equations, 59 (1985), 155-164.
doi: 10.1016/0022-0396(85)90152-4.![]() ![]() ![]() |
[14] |
H. Egger, H. W. Engl and M. V. Klibanov, Global uniqueness and Hölder stability for recovering a nonlinear source term in a parabolic equation, Inverse Problems, 21 (2005), 271-290.
doi: 10.1088/0266-5611/21/1/017.![]() ![]() ![]() |
[15] |
R. Glowinski, J.-L. Lions and J. He, Exact and Approximate Controllability for Distributed Parameter Systems, A Numerical Approach, Cambridge University Press, Cambridge, 2008.
doi: 10.1017/CBO9780511721595.![]() ![]() ![]() |
[16] |
M. Gonzalez-Burgos and L. de Teresa, Some results on controllability for linear and nonlinear heat equations in unbounded domains, Adv. Differential Equations, 12 (2007), 1201-1240.
![]() ![]() |
[17] |
O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, American Mathematical Society, Providence, R.I., 1968.
![]() ![]() |
[18] |
A. Lorenzi, An inverse problem for a semilinear parabolic equation, Ann. Mat. Pura Appl., 131 (1982), 145-166.
doi: 10.1007/BF01765150.![]() ![]() ![]() |
[19] |
A. Lunardi, Analytic Semigroups and Optima Regularity in Parabolic Problems, Modern Birkhauser Classics. Brikhauser/Springer Basel AG, Baser, 1995.
![]() ![]() |
[20] |
S.-I. Nakamura, A note on uniqueness in an inverse problem for a semilinear parabolic equation, Nihonkai Math. J., 12 (2001), 71-73.
![]() ![]() |
[21] |
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Vol. 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1.![]() ![]() ![]() |
[22] |
M. S. Pilant and W. Rundell, An inverse problem for a nonlinear parabolic equation, Comm. Partial Differential Equations, 11 (1986), 445-457.
doi: 10.1080/03605308608820430.![]() ![]() ![]() |
[23] |
L. Roques, M. D. Chekroun, M. Cristofol, S. Soubeyrand and M. Ghil, Parameter estimation for energy balance models with memory, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 470 (2014), 20140349, 20pp.
doi: 10.1098/rspa.2014.0349.![]() ![]() ![]() |
[24] |
L. Roques and M. Cristofol, On the determination of the nonlinearity from localized measurements in a reaction-diffusion equation, Nonlinearity, 23 (2010), 675-686.
![]() ![]() |
[25] |
L. Roques and M. Cristofol, The inverse problem of determining several coefficients in a nonlinear Lotka-Volterra system, Inverse Problems, 28 (2012), 075007, 12pp.
![]() ![]() |
[26] |
J.-C. Saut and B. Scheurer, Unique continuation for some evolution equations, J. Differential Equations, 66 (1987), 118-139.
doi: 10.1016/0022-0396(87)90043-X.![]() ![]() ![]() |
[27] |
M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Problems, 25 (2009), 123013, 75pp.
![]() ![]() |
[28] |
G. Yuan and M. Yamamoto, Lipschitz stability in the determination of the principal part of a parabolic equation, ESAIM : Control Optim. Calc. Var., 15 (2009), 525-554.
doi: 10.1051/cocv:2008043.![]() ![]() ![]() |
Representation of
Lighted lateral boundary from
A neighborhood of the lateral boundary