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September  2020, 10(3): 623-642. doi: 10.3934/mcrf.2020013

Controllability of a system of degenerate parabolic equations with non-diagonalizable diffusion matrix

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

* Corresponding author: fadilimed@live.fr

Received  October 2018 Revised  May 2019 Published  December 2019

In this paper we study the null controllability of some non diagonalizable degenerate parabolic systems of PDEs, we assume that the diffusion, coupling and controls matrices are constant and we characterize the null controllability by an algebraic condition so called Kalman's rank condition.

Citation: El Mustapha Ait Ben Hassi, Mohamed Fadili, Lahcen Maniar. Controllability of a system of degenerate parabolic equations with non-diagonalizable diffusion matrix. Mathematical Control & Related Fields, 2020, 10 (3) : 623-642. doi: 10.3934/mcrf.2020013
References:
[1]

E. M. Ait Ben HassiF. Ammar KhodjaA. Hajjaj and L. Maniar, Null controllability of degenerate parabolic cascade systems, Portugal. Math., 68 (2011), 345-367.  doi: 10.4171/PM/1895.  Google Scholar

[2]

E. M. Ait Ben HassiF. Ammar KhodjaA. 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.  Google Scholar

[3]

E. M. Ait Ben HassiM. Fadili and L. Maniar, On algebraic condition for null controllability of some coupled degenerate systems, Mathematical Control and Related Fields, 9 (2019), 77-95.  doi: 10.3934/mcrf.2019004.  Google Scholar

[4]

F. Alabau-BoussouiraP. Cannarsa and G. Fragnelli, Carleman estimates for degenerate parabolic operators with application to null controlability, J. Evol. Equ., 6 (2006), 161-204.  doi: 10.1007/s00028-006-0222-6.  Google Scholar

[5]

F. Ammar KhodjaA. BenabdallahM. González-Burgos and L. de Teresa, Recent results on the controllability of linear coupled parabolic problems: A survey, Mathematical Control and Related Fields, 1 (2011), 267-306.  doi: 10.3934/mcrf.2011.1.267.  Google Scholar

[6]

F. Ammar KhodjaA. BenabdallahC. Dupaix and M. González-Burgos, A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems, Diff. Equ. Appl., 1 (2009), 427-457.  doi: 10.7153/dea-01-24.  Google Scholar

[7]

F. Ammar KhodjaA. BenabdallahC. Dupaix and M. González-Burgos, A Kalman rank condition for the localized distributed controllability of a class of linear parabolic systems, J. Evol. Equ., 9 (2009), 267-291.  doi: 10.1007/s00028-009-0008-8.  Google Scholar

[8]

F. Ammar KhodjaA. Benabdellah and C. Dupaix, Null-controllability for some reaction-diffusion systems with one control force, J. Math. Anal. Appl., 320 (2006), 928-943.  doi: 10.1016/j.jmaa.2005.07.060.  Google Scholar

[9]

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

[10]

P. Cannarsa and G. Fragnelli, Null controllability of semilinear degenerate parabolic equations in bounded domains, Electronic Journal of Differential Equations, 2006 (2006), 20 pp.   Google Scholar

[11]

P. CannarsaP. Martinez and J. Vancostenoble, Null controllability of degenerate heat equations, Adv. Differential Equations, 10 (2005), 153-190.   Google Scholar

[12]

P. CannarsaP. Martinez and J. Vancostenoble, Carleman estimates for a class of degenerate parabolic operators, SIAM J. Control Optim., 47 (2008), 1-19.  doi: 10.1137/04062062X.  Google Scholar

[13]

P. CannarsaP. Martinez and J. Vancostenoble, Global Carleman estimates for degenerate parabolic operators with applications, Memoirs of the American Mathematical Society, 239 (2016).  doi: 10.1090/memo/1133.  Google Scholar

[14]

P. Cannarsa and L. de Teresa, Controllability of 1-D coupled degenerate parabolic equations, Electronic Journal of Differential Equations, 2009 (2009), 21 pp.   Google Scholar

[15]

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.  Google Scholar

[16]

E. Fernandez-CaraM. Gonzalez-Burgos and L. de Teresa, Controllability of linear and semilinear non-diagonalizable parabolic systems, ESAIM Control Optim. Calc. Var., 21 (2015), 1178-1204.  doi: 10.1051/cocv/2014063.  Google Scholar

[17]

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.  Google Scholar

[18]

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

[19]

M. Gueye, Exact boundary controllability of 1-D parabolic and hyperbolic degenerate equations, SIAM J. Control Optim., 52 (2014), 2037-2054.  doi: 10.1137/120901374.  Google Scholar

[20]

G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur, Comm. in PDE, 20 (1995), 335-356.  doi: 10.1080/03605309508821097.  Google Scholar

[21]

R. D. Meyer, Degenerate elliptic differential systems, J. Math. Anal. Appl., 29 (1970), 436-442.  doi: 10.1016/0022-247X(70)90093-4.  Google Scholar

[22]

L. A. F. de Oliveira, On reaction-diffusion systems, Electron. J. Differential Equations, 1998 (1998), 10 pp.   Google Scholar

[23]

J. Zabczyk, Mathematical Control Theory. An Introduction, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2008. doi: 10.1007/978-0-8176-4733-9.  Google Scholar

[24]

E. Zuazua, Exact controllability for semilinear wave equations in one space dimension, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 109-129.  doi: 10.1016/S0294-1449(16)30221-9.  Google Scholar

show all references

References:
[1]

E. M. Ait Ben HassiF. Ammar KhodjaA. Hajjaj and L. Maniar, Null controllability of degenerate parabolic cascade systems, Portugal. Math., 68 (2011), 345-367.  doi: 10.4171/PM/1895.  Google Scholar

[2]

E. M. Ait Ben HassiF. Ammar KhodjaA. 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.  Google Scholar

[3]

E. M. Ait Ben HassiM. Fadili and L. Maniar, On algebraic condition for null controllability of some coupled degenerate systems, Mathematical Control and Related Fields, 9 (2019), 77-95.  doi: 10.3934/mcrf.2019004.  Google Scholar

[4]

F. Alabau-BoussouiraP. Cannarsa and G. Fragnelli, Carleman estimates for degenerate parabolic operators with application to null controlability, J. Evol. Equ., 6 (2006), 161-204.  doi: 10.1007/s00028-006-0222-6.  Google Scholar

[5]

F. Ammar KhodjaA. BenabdallahM. González-Burgos and L. de Teresa, Recent results on the controllability of linear coupled parabolic problems: A survey, Mathematical Control and Related Fields, 1 (2011), 267-306.  doi: 10.3934/mcrf.2011.1.267.  Google Scholar

[6]

F. Ammar KhodjaA. BenabdallahC. Dupaix and M. González-Burgos, A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems, Diff. Equ. Appl., 1 (2009), 427-457.  doi: 10.7153/dea-01-24.  Google Scholar

[7]

F. Ammar KhodjaA. BenabdallahC. Dupaix and M. González-Burgos, A Kalman rank condition for the localized distributed controllability of a class of linear parabolic systems, J. Evol. Equ., 9 (2009), 267-291.  doi: 10.1007/s00028-009-0008-8.  Google Scholar

[8]

F. Ammar KhodjaA. Benabdellah and C. Dupaix, Null-controllability for some reaction-diffusion systems with one control force, J. Math. Anal. Appl., 320 (2006), 928-943.  doi: 10.1016/j.jmaa.2005.07.060.  Google Scholar

[9]

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

[10]

P. Cannarsa and G. Fragnelli, Null controllability of semilinear degenerate parabolic equations in bounded domains, Electronic Journal of Differential Equations, 2006 (2006), 20 pp.   Google Scholar

[11]

P. CannarsaP. Martinez and J. Vancostenoble, Null controllability of degenerate heat equations, Adv. Differential Equations, 10 (2005), 153-190.   Google Scholar

[12]

P. CannarsaP. Martinez and J. Vancostenoble, Carleman estimates for a class of degenerate parabolic operators, SIAM J. Control Optim., 47 (2008), 1-19.  doi: 10.1137/04062062X.  Google Scholar

[13]

P. CannarsaP. Martinez and J. Vancostenoble, Global Carleman estimates for degenerate parabolic operators with applications, Memoirs of the American Mathematical Society, 239 (2016).  doi: 10.1090/memo/1133.  Google Scholar

[14]

P. Cannarsa and L. de Teresa, Controllability of 1-D coupled degenerate parabolic equations, Electronic Journal of Differential Equations, 2009 (2009), 21 pp.   Google Scholar

[15]

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.  Google Scholar

[16]

E. Fernandez-CaraM. Gonzalez-Burgos and L. de Teresa, Controllability of linear and semilinear non-diagonalizable parabolic systems, ESAIM Control Optim. Calc. Var., 21 (2015), 1178-1204.  doi: 10.1051/cocv/2014063.  Google Scholar

[17]

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.  Google Scholar

[18]

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

[19]

M. Gueye, Exact boundary controllability of 1-D parabolic and hyperbolic degenerate equations, SIAM J. Control Optim., 52 (2014), 2037-2054.  doi: 10.1137/120901374.  Google Scholar

[20]

G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur, Comm. in PDE, 20 (1995), 335-356.  doi: 10.1080/03605309508821097.  Google Scholar

[21]

R. D. Meyer, Degenerate elliptic differential systems, J. Math. Anal. Appl., 29 (1970), 436-442.  doi: 10.1016/0022-247X(70)90093-4.  Google Scholar

[22]

L. A. F. de Oliveira, On reaction-diffusion systems, Electron. J. Differential Equations, 1998 (1998), 10 pp.   Google Scholar

[23]

J. Zabczyk, Mathematical Control Theory. An Introduction, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2008. doi: 10.1007/978-0-8176-4733-9.  Google Scholar

[24]

E. Zuazua, Exact controllability for semilinear wave equations in one space dimension, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 109-129.  doi: 10.1016/S0294-1449(16)30221-9.  Google Scholar

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