-
Previous Article
Stabilization of multidimensional wave equation with locally boundary fractional dissipation law under geometric conditions
- MCRF Home
- This Issue
-
Next Article
Robust optimal investment and reinsurance of an insurer under Jump-diffusion models
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 |
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.
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. |
[1] |
El Mustapha Ait Ben Hassi, Farid Ammar khodja, Abdelkarim Hajjaj, Lahcen Maniar. Carleman Estimates and null controllability of coupled degenerate systems. Evolution Equations and 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 and 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 and 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 and 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 and Related Fields, 2016, 6 (2) : 185-216. doi: 10.3934/mcrf.2016001 |
[6] |
J. Carmelo Flores, Luz De Teresa. Null controllability of one dimensional degenerate parabolic equations with first order terms. Discrete and Continuous Dynamical Systems - B, 2020, 25 (10) : 3963-3981. doi: 10.3934/dcdsb.2020136 |
[7] |
Brahim Allal, Abdelkarim Hajjaj, Lahcen Maniar, Jawad Salhi. Null controllability for singular cascade systems of $ n $-coupled degenerate parabolic equations by one control force. Evolution Equations and Control Theory, 2021, 10 (3) : 545-573. doi: 10.3934/eect.2020080 |
[8] |
Kuntal Bhandari, Franck Boyer. Boundary null-controllability of coupled parabolic systems with Robin conditions. Evolution Equations and Control Theory, 2021, 10 (1) : 61-102. doi: 10.3934/eect.2020052 |
[9] |
Farid Ammar Khodja, Cherif Bouzidi, Cédric Dupaix, Lahcen Maniar. Null controllability of retarded parabolic equations. Mathematical Control and Related Fields, 2014, 4 (1) : 1-15. doi: 10.3934/mcrf.2014.4.1 |
[10] |
Chunpeng Wang, Yanan Zhou, Runmei Du, Qiang Liu. Carleman estimate for solutions to a degenerate convection-diffusion equation. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4207-4222. doi: 10.3934/dcdsb.2018133 |
[11] |
R. Demarque, J. Límaco, L. Viana. Local null controllability of coupled degenerate systems with nonlocal terms and one control force. Evolution Equations and Control Theory, 2020, 9 (3) : 605-634. doi: 10.3934/eect.2020026 |
[12] |
Brahim Allal, Genni Fragnelli, Jawad Salhi*. Controllability for degenerate/singular parabolic systems involving memory terms. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022071 |
[13] |
Lianwen Wang. Approximate controllability and approximate null controllability of semilinear systems. Communications on Pure and Applied Analysis, 2006, 5 (4) : 953-962. doi: 10.3934/cpaa.2006.5.953 |
[14] |
Sergei Avdonin, Jeff Park, Luz de Teresa. The Kalman condition for the boundary controllability of coupled 1-d wave equations. Evolution Equations and Control Theory, 2020, 9 (1) : 255-273. doi: 10.3934/eect.2020005 |
[15] |
Lahcen Maniar, Martin Meyries, Roland Schnaubelt. Null controllability for parabolic equations with dynamic boundary conditions. Evolution Equations and Control Theory, 2017, 6 (3) : 381-407. doi: 10.3934/eect.2017020 |
[16] |
Lydia Ouaili. Minimal time of null controllability of two parabolic equations. Mathematical Control and Related Fields, 2020, 10 (1) : 89-112. doi: 10.3934/mcrf.2019031 |
[17] |
Larbi Berrahmoune. Null controllability for distributed systems with time-varying constraint and applications to parabolic-like equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 3275-3303. doi: 10.3934/dcdsb.2020062 |
[18] |
Damien Allonsius, Franck Boyer. Boundary null-controllability of semi-discrete coupled parabolic systems in some multi-dimensional geometries. Mathematical Control and Related Fields, 2020, 10 (2) : 217-256. doi: 10.3934/mcrf.2019037 |
[19] |
Fen-Fen Yang. Harnack inequality and gradient estimate for functional G-SDEs with degenerate noise. Probability, Uncertainty and Quantitative Risk, , () : -. doi: 10.3934/puqr.2022008 |
[20] |
Hamid Maarouf. Local Kalman rank condition for linear time varying systems. Mathematical Control and Related Fields, 2022, 12 (2) : 433-446. doi: 10.3934/mcrf.2021029 |
2021 Impact Factor: 1.141
Tools
Metrics
Other articles
by authors
[Back to Top]