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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  September 2020 Early access  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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[11]

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

[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.

[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.

[14]

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

[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.

[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.

[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.

[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.

[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.

[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.

[21]

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

[22]

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

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[11]

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

[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.

[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.

[14]

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

[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.

[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.

[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.

[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.

[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.

[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.

[21]

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

[22]

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

[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.

[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.

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