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Strict dissipativity analysis for classes of optimal control problems involving probability density functions
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 |
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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 and Related Fields,
2021, 11
(4)
: 965-985.
doi: 10.3934/mcrf.2020054
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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. |
| [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. |
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. |
| [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. |
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