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December  2021, 11(4): 965-985. doi: 10.3934/mcrf.2020054

A stability result for the diffusion coefficient of the heat operator defined on an unbounded guide

Laure Cardoulis , , Michel Cristofol and  Morgan Morancey

Aix Marseille Univ, CNRS, Centrale Marseille, I2M, Marseille, France

* Corresponding author

Received  January 2020 Revised  October 2020 Published  December 2021 Early access  December 2020

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.

Keywords: Coefficient inverse problem, stability estimate, heat equation, unbounded domain, Carleman estimate, approximate controllability in regular norms.
Mathematics Subject Classification: Primary: 35R30; Secondary: 35K05.
Citation: Laure Cardoulis, Michel Cristofol, Morgan Morancey. A stability result for the diffusion coefficient of the heat operator defined on an unbounded guide. Mathematical Control & Related Fields, 2021, 11 (4) : 965-985. doi: 10.3934/mcrf.2020054
References:
[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.   Google Scholar

[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.  Google Scholar

[18]

A. Lorenzi, An inverse problem for a semilinear parabolic equation, Ann. Mat. Pura Appl., 131 (1982), 145-166.  doi: 10.1007/BF01765150.  Google Scholar

[19]

A. Lunardi, Analytic Semigroups and Optima Regularity in Parabolic Problems, Modern Birkhauser Classics. Brikhauser/Springer Basel AG, Baser, 1995.  Google Scholar

[20]

S.-I. Nakamura, A note on uniqueness in an inverse problem for a semilinear parabolic equation, Nihonkai Math. J., 12 (2001), 71-73.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Problems, 25 (2009), 123013, 75pp.  Google Scholar

[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.  Google Scholar

show all references

References:
[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.   Google Scholar

[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.  Google Scholar

[18]

A. Lorenzi, An inverse problem for a semilinear parabolic equation, Ann. Mat. Pura Appl., 131 (1982), 145-166.  doi: 10.1007/BF01765150.  Google Scholar

[19]

A. Lunardi, Analytic Semigroups and Optima Regularity in Parabolic Problems, Modern Birkhauser Classics. Brikhauser/Springer Basel AG, Baser, 1995.  Google Scholar

[20]

S.-I. Nakamura, A note on uniqueness in an inverse problem for a semilinear parabolic equation, Nihonkai Math. J., 12 (2001), 71-73.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Problems, 25 (2009), 123013, 75pp.  Google Scholar

[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.  Google Scholar

Figure 1.  Representation of $ \Omega_l $ and $ \Omega_L $ in a 3D setting
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Figure 2.  Lighted lateral boundary from $ a $
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Figure 3.  A neighborhood of the lateral boundary
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