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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[19]

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

[20]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[19]

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

[20]

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

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