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September  2020, 10(3): 643-667. doi: 10.3934/mcrf.2020014

Lipschitz stability for some coupled degenerate parabolic systems with locally distributed observations of one component

Brahim Allal 1, , Abdelkarim Hajjaj 1, , Lahcen Maniar 2, and  Jawad Salhi 1,,

1. 

Faculté des Sciences et Techniques, Université Hassan 1er, Laboratoire MISI, B.P. 577, Settat 26000, Morocco
2. 

Département de Mathématiques, Faculté des Sciences Semlalia, LMDP, UMMISCO (IRD-UPMC), Université Cadi Ayyad, Marrakech 40000, B.P. 2390, Morocco

* Corresponding author: Jawad Salhi

Received  November 2018 Revised  July 2019 Published  September 2020 Early access  December 2019

This article presents an inverse source problem for a cascade system of $ n $ coupled degenerate parabolic equations. In particular, we prove stability and uniqueness results for the inverse problem of determining the source terms by observations in an arbitrary subdomain over a time interval of only one component and data of the $ n $ components at a fixed positive time $ T' $ over the whole spatial domain. The proof is based on the application of a Carleman estimate with a single observation acting on a subdomain.

Keywords: Carleman estimates, stability estimate, degenerate equations, coupled systems, inverse problem.
Mathematics Subject Classification: Primary: 35R30, 35K65; Secondary: 93B30.
Citation: Brahim Allal, Abdelkarim Hajjaj, Lahcen Maniar, Jawad Salhi. Lipschitz stability for some coupled degenerate parabolic systems with locally distributed observations of one component. Mathematical Control and Related Fields, 2020, 10 (3) : 643-667. doi: 10.3934/mcrf.2020014
References:
[1]

B. Ainseba, M. Bendahmane and Y. He, Stability of conductivities in an inverse problem in the reaction-diffusion system in electrocardiology, Netw. Heterog. Media, 10 (2015), 369-385.  doi: 10.3934/nhm.2015.10.369.

[2]

E. M. Ait Ben Hassi, F. Ammar Khodja, A. Hajjaj and L. Maniar, Carleman estimates and null controllability of coupled degenerate systems, Evol. Equ. Control Theory, 2 (2013), 441-459.  doi: 10.3934/eect.2013.2.441.

[3]

F. Alabau-Boussouira, P. Cannarsa and G. Fragnelli, Carleman estimates for degenerate parabolic operators with applications to null controllability, J. Evol. Equ., 6 (2006), 161-204.  doi: 10.1007/s00028-006-0222-6.

[4]

F. Alabau-Boussouira, P. Cannarsa and M. Yamamoto, Source reconstruction by partial measurements for a class of hyperbolic systems in cascade, Mathematical paradigms of climate science, Springer INdAM Ser., Springer, [Cham], 15 (2016), 35-50. 

[5]

M. Bellassoued and M. Yamamoto, Carleman Estimates and Applications to Inverse Problems for Hyperbolic Systems, Springer Monographs in Mathematics, Springer, Tokyo, 2017. doi: 10.1007/978-4-431-56600-7.

[6]

M. Bellassoued and M. Yamamoto, Carleman estimates and an inverse heat source problem for the thermoelasticity system, Inverse Problems, 27 (2011), 015006, 18 pp. doi: 10.1088/0266-5611/27/1/015006.

[7]

A. Benabdallah, M. Cristofol, P. Gaitan and M. Yamamoto, Inverse problem for a parabolic system with two components by measurements of one component, Appl. Anal., 88 (2009), 683-709.  doi: 10.1080/00036810802555490.

[8]

A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite-Dimensional Systems. Vol. 1, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1992.

[9]

I. Boutaayamou, G. Fragnelli and L. Maniar, Inverse problems for parabolic equations with interior degeneracy and Neumann boundary conditions, J. Inverse III-Posed Probl, 24 (2016), 275-292.  doi: 10.1515/jiip-2014-0032.

[10]

I. Boutaayamou, G. Fragnelli and L. Maniar, Lipschitz stability for linear parabolic systems with interior degeneracy, Electron. J. Differential Equations, 2014 (2014), 26 pp.

[11]

I. Boutaayamou, A. Hajjaj and L. Maniar, Lipschitz stability for degenerate parabolic systems, Electron. J. Differential Equations, 2014 (2014), 15 pp.

[12]

A. L. Bukhgeim and M. V. Klibanov, Global uniqueness of a class of multidimensional inverse problems, Soviet Math. Dokl., 24 (1981), 244-247. 

[13]

M. Campiti, G. Metafune and D. Pallara, Degenerate self-adjoint evolution equations on the unit interval, Semigroup Forum, 57 (1998), 1-36.  doi: 10.1007/PL00005959.

[14]

P. Cannarsa, P. Martinez and J. Vancostenoble, Global Carleman estimates for degenerate parabolic operators with applications, Mem. Amer. Math. Soc., 239 (2016). doi: 10.1090/memo/1133.

[15]

P. Cannarsa, J. Tort and M. Yamamoto, Determination of source terms in a degenerate parabolic equation, Inverse Problems, 26 (2010), 105003, 20 pp. doi: 10.1088/0266-5611/26/10/105003.

[16]

M. Cristofol, P. Gaitan, K. Niinimäki and O. Poisson, Inverse problem for a coupled parabolic system with discontinuous conductivities: One-dimensional case, Inverse Problems and Imaging, 7 (2013), 159-182.  doi: 10.3934/ipi.2013.7.159.

[17]

M. Cristofol, P. Gaitan and H. Ramoul, Inverse problems for a $2\times2$ reaction-diffusion system using a Carleman estimate with one observation, Inverse Problems, 22 (2006), 1561-1573.  doi: 10.1088/0266-5611/22/5/003.

[18]

M. Cristofol, P. Gaitan, H. Ramoul and M. Yamamoto, Identification of two independent coefficients with one observation for a nonlinear parabolic system, Appl. Anal., 91 (2012), 2073-2081.  doi: 10.1080/00036811.2011.583240.

[19]

V. Dinakar, N. B. Balan and K. Balachandran, Identification of source terms in a coupled age-structured population model with discontinuous diffusion coefficients, AIMS Mathematics, 2 (2017), 81-95.  doi: 10.3934/Math.2017.1.81.

[20]

M. Fadili and L. Maniar, Null controllability of $n$-coupled degenerate parabolic systems with $m$-controls, J. Evol. Equ., 17 (2017), 1311-1340.  doi: 10.1007/s00028-017-0385-3.

[21]

A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, 34. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.

[22]

M. Gonzalez-Burgos and L. de Teresa, Controllability results for cascade systems of $m$ coupled parabolic PDEs by one control force, Portugal. Math., 67 (2010), 91-113.  doi: 10.4171/PM/1859.

[23]

O. Y. Imanuvilov and M. Yamamoto, Lipschitz stability in inverse parabolic problem by Carleman estimates, Inverse Problems, 14 (1998), 1229-1245.  doi: 10.1088/0266-5611/14/5/009.

[24]

L. Roques and M. Cristofol, The inverse problem of determining several coefficients in a nonlinear Lotka-Volterra system, Inverse Problems, 28 (2012), 075007, 12 pp. doi: 10.1088/0266-5611/28/7/075007.

[25]

J. Tort, An inverse diffusion problem in a degenerate parabolic equation, Monografias, Real Academia de Ciencias de Zaragoza, 38 (2012), 137-145. 

[26]

J. Tort and J. Vancostenoble, Determination of the insolation function in the nonlinear Sellers climate model, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 683-713.  doi: 10.1016/j.anihpc.2012.03.003.

[27]

J. Vancostenoble, Lipschitz stability in inverse source problems for singular parabolic equations, Communications in Partial Differential Equations, 36 (2011), 1287-1317.  doi: 10.1080/03605302.2011.587491.

[28]

B. Wu and J. Yu, Hölder stability of an inverse problem for a strongly coupled reaction-diffusion system, IMA J. Appl. Math., 82 (2017), 424-444.  doi: 10.1093/imamat/hxw058.

[29]

M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Problems, 25 (2009), 123013, 75 pp. doi: 10.1088/0266-5611/25/12/123013.

show all references

References:
[1]

B. Ainseba, M. Bendahmane and Y. He, Stability of conductivities in an inverse problem in the reaction-diffusion system in electrocardiology, Netw. Heterog. Media, 10 (2015), 369-385.  doi: 10.3934/nhm.2015.10.369.

[2]

E. M. Ait Ben Hassi, F. Ammar Khodja, A. Hajjaj and L. Maniar, Carleman estimates and null controllability of coupled degenerate systems, Evol. Equ. Control Theory, 2 (2013), 441-459.  doi: 10.3934/eect.2013.2.441.

[3]

F. Alabau-Boussouira, P. Cannarsa and G. Fragnelli, Carleman estimates for degenerate parabolic operators with applications to null controllability, J. Evol. Equ., 6 (2006), 161-204.  doi: 10.1007/s00028-006-0222-6.

[4]

F. Alabau-Boussouira, P. Cannarsa and M. Yamamoto, Source reconstruction by partial measurements for a class of hyperbolic systems in cascade, Mathematical paradigms of climate science, Springer INdAM Ser., Springer, [Cham], 15 (2016), 35-50. 

[5]

M. Bellassoued and M. Yamamoto, Carleman Estimates and Applications to Inverse Problems for Hyperbolic Systems, Springer Monographs in Mathematics, Springer, Tokyo, 2017. doi: 10.1007/978-4-431-56600-7.

[6]

M. Bellassoued and M. Yamamoto, Carleman estimates and an inverse heat source problem for the thermoelasticity system, Inverse Problems, 27 (2011), 015006, 18 pp. doi: 10.1088/0266-5611/27/1/015006.

[7]

A. Benabdallah, M. Cristofol, P. Gaitan and M. Yamamoto, Inverse problem for a parabolic system with two components by measurements of one component, Appl. Anal., 88 (2009), 683-709.  doi: 10.1080/00036810802555490.

[8]

A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite-Dimensional Systems. Vol. 1, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1992.

[9]

I. Boutaayamou, G. Fragnelli and L. Maniar, Inverse problems for parabolic equations with interior degeneracy and Neumann boundary conditions, J. Inverse III-Posed Probl, 24 (2016), 275-292.  doi: 10.1515/jiip-2014-0032.

[10]

I. Boutaayamou, G. Fragnelli and L. Maniar, Lipschitz stability for linear parabolic systems with interior degeneracy, Electron. J. Differential Equations, 2014 (2014), 26 pp.

[11]

I. Boutaayamou, A. Hajjaj and L. Maniar, Lipschitz stability for degenerate parabolic systems, Electron. J. Differential Equations, 2014 (2014), 15 pp.

[12]

A. L. Bukhgeim and M. V. Klibanov, Global uniqueness of a class of multidimensional inverse problems, Soviet Math. Dokl., 24 (1981), 244-247. 

[13]

M. Campiti, G. Metafune and D. Pallara, Degenerate self-adjoint evolution equations on the unit interval, Semigroup Forum, 57 (1998), 1-36.  doi: 10.1007/PL00005959.

[14]

P. Cannarsa, P. Martinez and J. Vancostenoble, Global Carleman estimates for degenerate parabolic operators with applications, Mem. Amer. Math. Soc., 239 (2016). doi: 10.1090/memo/1133.

[15]

P. Cannarsa, J. Tort and M. Yamamoto, Determination of source terms in a degenerate parabolic equation, Inverse Problems, 26 (2010), 105003, 20 pp. doi: 10.1088/0266-5611/26/10/105003.

[16]

M. Cristofol, P. Gaitan, K. Niinimäki and O. Poisson, Inverse problem for a coupled parabolic system with discontinuous conductivities: One-dimensional case, Inverse Problems and Imaging, 7 (2013), 159-182.  doi: 10.3934/ipi.2013.7.159.

[17]

M. Cristofol, P. Gaitan and H. Ramoul, Inverse problems for a $2\times2$ reaction-diffusion system using a Carleman estimate with one observation, Inverse Problems, 22 (2006), 1561-1573.  doi: 10.1088/0266-5611/22/5/003.

[18]

M. Cristofol, P. Gaitan, H. Ramoul and M. Yamamoto, Identification of two independent coefficients with one observation for a nonlinear parabolic system, Appl. Anal., 91 (2012), 2073-2081.  doi: 10.1080/00036811.2011.583240.

[19]

V. Dinakar, N. B. Balan and K. Balachandran, Identification of source terms in a coupled age-structured population model with discontinuous diffusion coefficients, AIMS Mathematics, 2 (2017), 81-95.  doi: 10.3934/Math.2017.1.81.

[20]

M. Fadili and L. Maniar, Null controllability of $n$-coupled degenerate parabolic systems with $m$-controls, J. Evol. Equ., 17 (2017), 1311-1340.  doi: 10.1007/s00028-017-0385-3.

[21]

A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, 34. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.

[22]

M. Gonzalez-Burgos and L. de Teresa, Controllability results for cascade systems of $m$ coupled parabolic PDEs by one control force, Portugal. Math., 67 (2010), 91-113.  doi: 10.4171/PM/1859.

[23]

O. Y. Imanuvilov and M. Yamamoto, Lipschitz stability in inverse parabolic problem by Carleman estimates, Inverse Problems, 14 (1998), 1229-1245.  doi: 10.1088/0266-5611/14/5/009.

[24]

L. Roques and M. Cristofol, The inverse problem of determining several coefficients in a nonlinear Lotka-Volterra system, Inverse Problems, 28 (2012), 075007, 12 pp. doi: 10.1088/0266-5611/28/7/075007.

[25]

J. Tort, An inverse diffusion problem in a degenerate parabolic equation, Monografias, Real Academia de Ciencias de Zaragoza, 38 (2012), 137-145. 

[26]

J. Tort and J. Vancostenoble, Determination of the insolation function in the nonlinear Sellers climate model, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 683-713.  doi: 10.1016/j.anihpc.2012.03.003.

[27]

J. Vancostenoble, Lipschitz stability in inverse source problems for singular parabolic equations, Communications in Partial Differential Equations, 36 (2011), 1287-1317.  doi: 10.1080/03605302.2011.587491.

[28]

B. Wu and J. Yu, Hölder stability of an inverse problem for a strongly coupled reaction-diffusion system, IMA J. Appl. Math., 82 (2017), 424-444.  doi: 10.1093/imamat/hxw058.

[29]

M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Problems, 25 (2009), 123013, 75 pp. doi: 10.1088/0266-5611/25/12/123013.

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