• Previous Article
    Optimal control of the coefficient for the regional fractional $p$-Laplace equation: Approximation and convergence
  • MCRF Home
  • This Issue
  • Next Article
    Stabilization of multidimensional wave equation with locally boundary fractional dissipation law under geometric conditions
doi: 10.3934/mcrf.2019004

On Algebraic condition for null controllability of some coupled degenerate systems

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.

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, doi: 10.3934/mcrf.2019004
References:
[1]

E. M. Ait BenhassiF. 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 BenhassiF. 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]

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.

[4]

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.

[5]

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.

[6]

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

[10]

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.

[11]

P. CannarsaP. 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 BenhassiF. 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 BenhassiF. 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]

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.

[4]

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.

[5]

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.

[6]

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

[10]

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.

[11]

P. CannarsaP. 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.

[1]

El Mustapha Ait Ben Hassi, Farid Ammar khodja, Abdelkarim Hajjaj, Lahcen Maniar. Carleman Estimates and null controllability of coupled degenerate systems. Evolution Equations & Control Theory, 2013, 2 (3) : 441-459. doi: 10.3934/eect.2013.2.441

[2]

Genni Fragnelli. Null controllability of degenerate parabolic equations in non divergence form via Carleman estimates. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 687-701. doi: 10.3934/dcdss.2013.6.687

[3]

Piermarco Cannarsa, Genni Fragnelli, Dario Rocchetti. Null controllability of degenerate parabolic operators with drift. Networks & Heterogeneous Media, 2007, 2 (4) : 695-715. doi: 10.3934/nhm.2007.2.695

[4]

Lingyang Liu, Xu Liu. Controllability and observability of some coupled stochastic parabolic systems. Mathematical Control & Related Fields, 2018, 8 (3&4) : 829-854. doi: 10.3934/mcrf.2018037

[5]

Farid Ammar Khodja, Franz Chouly, Michel Duprez. Partial null controllability of parabolic linear systems. Mathematical Control & Related Fields, 2016, 6 (2) : 185-216. doi: 10.3934/mcrf.2016001

[6]

Farid Ammar Khodja, Cherif Bouzidi, Cédric Dupaix, Lahcen Maniar. Null controllability of retarded parabolic equations. Mathematical Control & Related Fields, 2014, 4 (1) : 1-15. doi: 10.3934/mcrf.2014.4.1

[7]

Lianwen Wang. Approximate controllability and approximate null controllability of semilinear systems. Communications on Pure & Applied Analysis, 2006, 5 (4) : 953-962. doi: 10.3934/cpaa.2006.5.953

[8]

Chunpeng Wang, Yanan Zhou, Runmei Du, Qiang Liu. Carleman estimate for solutions to a degenerate convection-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2018, 22 (11) : 1-16. doi: 10.3934/dcdsb.2018133

[9]

Lahcen Maniar, Martin Meyries, Roland Schnaubelt. Null controllability for parabolic equations with dynamic boundary conditions. Evolution Equations & Control Theory, 2017, 6 (3) : 381-407. doi: 10.3934/eect.2017020

[10]

Enrique Fernández-Cara, Luz de Teresa. Null controllability of a cascade system of parabolic-hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 699-714. doi: 10.3934/dcds.2004.11.699

[11]

Piermarco Cannarsa, Patrick Martinez, Judith Vancostenoble. Persistent regional null contrillability for a class of degenerate parabolic equations. Communications on Pure & Applied Analysis, 2004, 3 (4) : 607-635. doi: 10.3934/cpaa.2004.3.607

[12]

Thuy N. T. Nguyen. Carleman estimates for semi-discrete parabolic operators with a discontinuous diffusion coefficient and applications to controllability. Mathematical Control & Related Fields, 2014, 4 (2) : 203-259. doi: 10.3934/mcrf.2014.4.203

[13]

Tatsien Li, Bopeng Rao, Zhiqiang Wang. Exact boundary controllability and observability for first order quasilinear hyperbolic systems with a kind of nonlocal boundary conditions. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 243-257. doi: 10.3934/dcds.2010.28.243

[14]

Enrique Fernández-Cara, Manuel González-Burgos, Luz de Teresa. Null-exact controllability of a semilinear cascade system of parabolic-hyperbolic equations. Communications on Pure & Applied Analysis, 2006, 5 (3) : 639-658. doi: 10.3934/cpaa.2006.5.639

[15]

Judith Vancostenoble. Improved Hardy-Poincaré inequalities and sharp Carleman estimates for degenerate/singular parabolic problems. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 761-790. doi: 10.3934/dcdss.2011.4.761

[16]

Morteza Fotouhi, Leila Salimi. Controllability results for a class of one dimensional degenerate/singular parabolic equations. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1415-1430. doi: 10.3934/cpaa.2013.12.1415

[17]

Peng Gao. Global Carleman estimate for the Kawahara equation and its applications. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1853-1874. doi: 10.3934/cpaa.2018088

[18]

Sun-Sig Byun, Yunsoo Jang. Calderón-Zygmund estimate for homogenization of parabolic systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6689-6714. doi: 10.3934/dcds.2016091

[19]

Felipe Wallison Chaves-Silva, Sergio Guerrero, Jean Pierre Puel. Controllability of fast diffusion coupled parabolic systems. Mathematical Control & Related Fields, 2014, 4 (4) : 465-479. doi: 10.3934/mcrf.2014.4.465

[20]

Guillaume Olive. Boundary approximate controllability of some linear parabolic systems. Evolution Equations & Control Theory, 2014, 3 (1) : 167-189. doi: 10.3934/eect.2014.3.167

2017 Impact Factor: 0.631

Metrics

  • PDF downloads (13)
  • HTML views (70)
  • Cited by (0)

[Back to Top]