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March  2019, 9(1): 77-95. doi: 10.3934/mcrf.2019004

On Algebraic condition for null controllability of some coupled degenerate systems

Ait Ben Hassi El Mustapha , Fadili Mohamed , and  Maniar Lahcen

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

Dedicated to Professor H. Bouslous on the occasion of his 65th birthday

Received  September 2017 Revised  February 2018 Published  August 2018

In this paper we will generalize the Kalman rank condition for the null controllability to $n$-coupled linear degenerate parabolic systems with constant coefficients, diagonalizable diffusion matrix, and $m$-controls. For that we prove a global Carleman estimate for the solutions of a scalar $2n$-order parabolic equation then we infer from it an observability inequality for the corresponding adjoint system, and thus the null controllability.

Keywords: Parabolic degenerate systems, Carleman estimate, null controllability, observability inequality, Kalman condition.
Mathematics Subject Classification: 35K20, 35K65, 47D06, 93B05, 93B07.
Citation: Ait Ben Hassi El Mustapha, Fadili Mohamed, Maniar Lahcen. On Algebraic condition for null controllability of some coupled degenerate systems. Mathematical Control and Related Fields, 2019, 9 (1) : 77-95. doi: 10.3934/mcrf.2019004
References:
[1]

E. M. Ait Benhassi, F. Ammar Khodja, A. 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 Benhassi, 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. Ammar-Khodja, A. Benabdallah, M. 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.

[4]

F. Ammar-Khodja, A. Benabdallah, C. 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.

[5]

F. Ammar-Khodja, A. Benabdallah, C. 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.

[6]

F. Alabau-Boussouira, P. Cannarsa and G. Fragnelli, Carleman estimates for degenerate parabolic operators with application to nullcontrolability, J. evol. equ., 6 (2006), 161-204.  doi: 10.1007/s00028-006-0222-6.

[7]

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.

[8]

P. Cannarsa and G. Fragnelli, Null controllability of semilinear degenerate parabolic equations in bounded domains, Electronic Journal of Differential Equations, 136 (2006), 1-20. 

[9]

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

[10]

P. Cannarsa, P. 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.

[11]

P. Cannarsa, P. Martinez and J. Vancostenoble, Global Carleman estimates for degenerate parabolic operators with applications, Memoirs of the American Mathematical Society, 239 (2016), ⅸ+209 pp.  doi: 10.1090/memo/1133.

[12]

P. Cannarsa and L. de Teresa, Controllability of 1-d coupled degenerate parabolic equations, Electronic Journal of Differential Equations, 73 (2009), 1-21. 

[13]

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.

[14]

A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, Lectures notes series 34, Seoul National University Research Center, Seoul, 1996.

[15]

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.

[16]

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.

[17]

A. Hajjaj, Estimations de Carleman et Applications à la contrôolabilité à Zéro D'une Classe De Systèmes Paraboliques Dégénérés, Thèse d'Etat, Marrakech, 2013.

[18]

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

[19]

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

[20]

J. Zabczyk, Mathematical Control Theory, Birkhäuser, Boston, 1995. doi: 10.1007/978-0-8176-4733-9.

show all references

References:
[1]

E. M. Ait Benhassi, F. Ammar Khodja, A. 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 Benhassi, 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. Ammar-Khodja, A. Benabdallah, M. 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.

[4]

F. Ammar-Khodja, A. Benabdallah, C. 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.

[5]

F. Ammar-Khodja, A. Benabdallah, C. 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.

[6]

F. Alabau-Boussouira, P. Cannarsa and G. Fragnelli, Carleman estimates for degenerate parabolic operators with application to nullcontrolability, J. evol. equ., 6 (2006), 161-204.  doi: 10.1007/s00028-006-0222-6.

[7]

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.

[8]

P. Cannarsa and G. Fragnelli, Null controllability of semilinear degenerate parabolic equations in bounded domains, Electronic Journal of Differential Equations, 136 (2006), 1-20. 

[9]

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

[10]

P. Cannarsa, P. 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.

[11]

P. Cannarsa, P. Martinez and J. Vancostenoble, Global Carleman estimates for degenerate parabolic operators with applications, Memoirs of the American Mathematical Society, 239 (2016), ⅸ+209 pp.  doi: 10.1090/memo/1133.

[12]

P. Cannarsa and L. de Teresa, Controllability of 1-d coupled degenerate parabolic equations, Electronic Journal of Differential Equations, 73 (2009), 1-21. 

[13]

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.

[14]

A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, Lectures notes series 34, Seoul National University Research Center, Seoul, 1996.

[15]

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.

[16]

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.

[17]

A. Hajjaj, Estimations de Carleman et Applications à la contrôolabilité à Zéro D'une Classe De Systèmes Paraboliques Dégénérés, Thèse d'Etat, Marrakech, 2013.

[18]

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

[19]

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

[20]

J. Zabczyk, Mathematical Control Theory, Birkhäuser, Boston, 1995. doi: 10.1007/978-0-8176-4733-9.

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