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Controllability of a system of degenerate parabolic equations with non-diagonalizable diffusion matrix
Lipschitz stability for some coupled degenerate parabolic systems with locally distributed observations of one component
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 |
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.
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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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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