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

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.

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 and Related Fields, 2021, 11 (4) : 965-985. doi: 10.3934/mcrf.2020054
References:
[1]

A. BenabdallahM. CristofolP. 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. BellassouedO. 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. ChoulliE. 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. CristofolJ. GarnierF. 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. CristofolS. 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. EggerH. 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. GlowinskiJ.-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.

show all references

References:
[1]

A. BenabdallahM. CristofolP. 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. BellassouedO. 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. ChoulliE. 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. CristofolJ. GarnierF. 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. CristofolS. 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. EggerH. 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. GlowinskiJ.-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.

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