September  2013, 2(3): 441-459. doi: 10.3934/eect.2013.2.441

Carleman Estimates and null controllability of coupled degenerate systems

1. 

Département de Mathématiques, Faculté des Sciences Semlalia, LMDP, UMMISCO (IRD-UPMC), Université Cadi Ayyad, Marrakech, 40000, B.P. 2390,, Morocco

2. 

Laboratoire de Mathématiques de Besançon, UMR CNRS 6623, Université de Franche-Comté, 25030 Besançon Cedex, France

3. 

Département de Mathématiques et Informatique, Faculté des Sciences et Techniques, Labo. MISI, Université Hassan 1er Settat 26000, B.P. 577, Morocco

4. 

Département de Mathématiques, Faculté des Sciences Semlalia, Université Cadi Ayyad, Marrakech 40000, B.P. 2390, Morocco

Received  May 2012 Revised  June 2013 Published  July 2013

In this paper, we study the null controllability of weakly degenerate parabolic systems with two different diffusion coefficients and one control force. To obtain this aim, we had to develop new global Carleman estimates for a degenerate parabolic equation, with weight functions different from the ones of [2], [10] and [31].
Citation: 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
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. doi: 10.4171/PM/1895. Google Scholar

[2]

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

[3]

F. Ammar Khodja, A. Benabdellah and C. Dupaix, Null-controllability for some reaction-diffusion systems with one control force,, J. Math. Anal. Appl., 320 (2006), 928. doi: 10.1016/j.jmaa.2005.07.060. Google Scholar

[4]

F. Ammar Khodja, A. Benabdallah, C. Dupaix and I. Kostin, Controllability to the trajectories of phase-field models by one control force,, SIAM J. Control Optim., 42 (2003), 1661. doi: 10.1137/S0363012902417826. Google Scholar

[5]

F. Ammar Khodja, A. Benabdallah, C. Dupaix and I. Kostin, Null-controllability of some systems of parabolic type by one control force,, ESAIM Control Optim. Calc. Var., 11 (2005), 426. doi: 10.1051/cocv:2005013. Google Scholar

[6]

V. R. Cabanillas, S. B. Menezes and E. Zuazua, Null controllability in unbounded domains for the semilinear heat equation with nonlinearities involving gradient terms,, J. Optimization Theory and Applications, 110 (2001), 245. doi: 10.1023/A:1017515027783. Google Scholar

[7]

M. Campiti, G. Metafune and D. Pallara, Degenerate self-adjoint evolution equations on the unit interval,, Semigroup Forum, 57 (1998), 1. doi: 10.1007/PL00005959. Google Scholar

[8]

P. Cannarsa and L. De Teresa, Controllability of 1-d coupled degenerate parabolic equations,, Electron. J. Differential Equations, (): 1. Google Scholar

[9]

P. Cannarsa and G. Fragnelli, Null controllability of semilinear degenerate parabolic equations in bounded domains,, Electron. J. Differential Equations, (): 1. Google Scholar

[10]

P. Cannarsa, P. Martinez and J. Vancostenoble, Carleman estimates for a class of degenerate parabolic operators,, SIAM, 47 (2008), 1. doi: 10.1137/04062062X. Google Scholar

[11]

P. Cannarsa, G. Fragnelli and J. Vancostenoble, Regional controllability of semilinear degenerate parabolic equations in bounded domains,, J. Math. Anal. Appl., 320 (2006), 804. doi: 10.1016/j.jmaa.2005.07.006. Google Scholar

[12]

P. Cannarsa, P. Martinez and J. Vancostenoble, Persistent regional null controllability for a class of degenerate parabolic equations,, Commun. Pure Appl. Anal., 3 (2004), 607. doi: 10.3934/cpaa.2004.3.607. Google Scholar

[13]

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

[14]

P. Cannarsa, J. Tort and M. Yamamoto, Unique continuation and approximate controllability for a degenerate parabolic equation,, Appl. Anal., 91 (2012), 1409. doi: 10.1080/00036811.2011.639766. Google Scholar

[15]

L. De Teresa, Insensitizing controls for a semilinear heat equation,, Comm. Partial Differential Equations, 25 (2000), 39. doi: 10.1080/03605300008821507. Google Scholar

[16]

L. De Teresa and E. Zuazua, Approximate controllability of the semilinear heat equation in unbounded domains,, Nonlinear Analysis TMA, 37 (1999), 1059. doi: 10.1016/S0362-546X(98)00085-6. Google Scholar

[17]

K. J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Quations,", Springer-Verlag, (2000). Google Scholar

[18]

C. Fabre, J. P. Puel and E. Zuazua, Approximate controllability of the semilinear heat equation,, Proceedings of the Royal Society of Edinburg, 125 (1995), 31. doi: 10.1017/S0308210500030742. Google Scholar

[19]

H. O. Fattorini and D. L. Russell, Exact controllability theorems for linear parabolic equations in one space dimension,, Arch. Rat. Mech. Anal., 43 (1971), 272. Google Scholar

[20]

H. O. Fattorini and D. L. Russell, Uniform bounds on biorthogonal functions for real exponentials with an application to control theory of parabolic equations,, Quarterly of applied Maths, 32 (1974), 45. Google Scholar

[21]

E. Fernandez-Cara, Null controllability of the semilinear heat equation,, ESAIM Control Optim. Calc. Var., 2 (1997), 87. doi: 10.1051/cocv:1997104. Google Scholar

[22]

E. Fernandez-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations,, Annales de l'Institut Henry Poincaré, 17 (2000), 583. doi: 10.1016/S0294-1449(00)00117-7. Google Scholar

[23]

A. V. Fursikov and O. Y. Imanuvilov, "Controllability of Evolution Equations,", Lectures Notes Series 34, (1996). Google Scholar

[24]

M. Gonzalez-Burgos and L. de Teresa, Controllability results for cascade systems of m coupled parabolic PDEs by one control force,, Portugal. Math., 67 (2010), 91. doi: 10.4171/PM/1859. Google Scholar

[25]

M. Gonzalez-Burgos and R. Perez-Garcia, Controllability of some coupled parabolic systems by one control force,, C. R. Math. Acad. Sci. Paris, 340 (2005), 125. doi: 10.1016/j.crma.2004.11.025. Google Scholar

[26]

M. Gonzalez-Burgos and R. Perez-Garcia, Controllability results for some nonlinear coupled parabolic systems by one control force,, Asymptot. Anal., 46 (2006), 123. Google Scholar

[27]

S. Guerrero, Null controllability of some systems of two parabolic equations with one control force,, SIAM J. Control Optim., 46 (2007), 379. doi: 10.1137/060653135. Google Scholar

[28]

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

[29]

X. Liu, H. Gao and P. Lin, Null controllability of a cascade system of degenerate parabolic equations,, Acta Math. Sci. Ser. A Chin. Ed., 28 (2008), 985. Google Scholar

[30]

A. Lopez, X. Zhang and E. Zuazua, Null controllability of the heat equation as singular limit of the exact controllability of dissipative wave equations,, J. Math. Pures Appl., 79 (2000), 741. doi: 10.1016/S0021-7824(99)00144-0. Google Scholar

[31]

P. Martinez and J. Vancostenoble, Carleman estimates for one-dimensional degenerate heat equations,, J. Evol. Equ., 6 (2006), 325. doi: 10.1007/s00028-006-0214-6. Google Scholar

[32]

S. Micu and E. Zuazua, On the lack of null controllability of the heat equation on the half-line,, Trans. Amer. Math. Soc., 353 (2001), 1635. doi: 10.1090/S0002-9947-00-02665-9. Google Scholar

[33]

B. Opic and A. Kufner, "Hardy-Type Inequalities,", Longman Scientific and Technical, (1990). Google Scholar

[34]

D. L. Russell, A unified boundary controllability theory for hyperbolic and parabolic partial differential equations,, Studies in Applied Mathematics, 52 (1973), 189. Google Scholar

[35]

D. Tataru, A priori estimates of Carleman's type in domains with boundary,, J. Math. Pures et Appliquées, 73 (1994), 355. Google Scholar

[36]

E. Zuazua, Approximate controllability for semilinear heat equations with globally Lipschitz nonlinearites,, Control Cybernet., 28 (1999), 665. 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. doi: 10.4171/PM/1895. Google Scholar

[2]

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

[3]

F. Ammar Khodja, A. Benabdellah and C. Dupaix, Null-controllability for some reaction-diffusion systems with one control force,, J. Math. Anal. Appl., 320 (2006), 928. doi: 10.1016/j.jmaa.2005.07.060. Google Scholar

[4]

F. Ammar Khodja, A. Benabdallah, C. Dupaix and I. Kostin, Controllability to the trajectories of phase-field models by one control force,, SIAM J. Control Optim., 42 (2003), 1661. doi: 10.1137/S0363012902417826. Google Scholar

[5]

F. Ammar Khodja, A. Benabdallah, C. Dupaix and I. Kostin, Null-controllability of some systems of parabolic type by one control force,, ESAIM Control Optim. Calc. Var., 11 (2005), 426. doi: 10.1051/cocv:2005013. Google Scholar

[6]

V. R. Cabanillas, S. B. Menezes and E. Zuazua, Null controllability in unbounded domains for the semilinear heat equation with nonlinearities involving gradient terms,, J. Optimization Theory and Applications, 110 (2001), 245. doi: 10.1023/A:1017515027783. Google Scholar

[7]

M. Campiti, G. Metafune and D. Pallara, Degenerate self-adjoint evolution equations on the unit interval,, Semigroup Forum, 57 (1998), 1. doi: 10.1007/PL00005959. Google Scholar

[8]

P. Cannarsa and L. De Teresa, Controllability of 1-d coupled degenerate parabolic equations,, Electron. J. Differential Equations, (): 1. Google Scholar

[9]

P. Cannarsa and G. Fragnelli, Null controllability of semilinear degenerate parabolic equations in bounded domains,, Electron. J. Differential Equations, (): 1. Google Scholar

[10]

P. Cannarsa, P. Martinez and J. Vancostenoble, Carleman estimates for a class of degenerate parabolic operators,, SIAM, 47 (2008), 1. doi: 10.1137/04062062X. Google Scholar

[11]

P. Cannarsa, G. Fragnelli and J. Vancostenoble, Regional controllability of semilinear degenerate parabolic equations in bounded domains,, J. Math. Anal. Appl., 320 (2006), 804. doi: 10.1016/j.jmaa.2005.07.006. Google Scholar

[12]

P. Cannarsa, P. Martinez and J. Vancostenoble, Persistent regional null controllability for a class of degenerate parabolic equations,, Commun. Pure Appl. Anal., 3 (2004), 607. doi: 10.3934/cpaa.2004.3.607. Google Scholar

[13]

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

[14]

P. Cannarsa, J. Tort and M. Yamamoto, Unique continuation and approximate controllability for a degenerate parabolic equation,, Appl. Anal., 91 (2012), 1409. doi: 10.1080/00036811.2011.639766. Google Scholar

[15]

L. De Teresa, Insensitizing controls for a semilinear heat equation,, Comm. Partial Differential Equations, 25 (2000), 39. doi: 10.1080/03605300008821507. Google Scholar

[16]

L. De Teresa and E. Zuazua, Approximate controllability of the semilinear heat equation in unbounded domains,, Nonlinear Analysis TMA, 37 (1999), 1059. doi: 10.1016/S0362-546X(98)00085-6. Google Scholar

[17]

K. J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Quations,", Springer-Verlag, (2000). Google Scholar

[18]

C. Fabre, J. P. Puel and E. Zuazua, Approximate controllability of the semilinear heat equation,, Proceedings of the Royal Society of Edinburg, 125 (1995), 31. doi: 10.1017/S0308210500030742. Google Scholar

[19]

H. O. Fattorini and D. L. Russell, Exact controllability theorems for linear parabolic equations in one space dimension,, Arch. Rat. Mech. Anal., 43 (1971), 272. Google Scholar

[20]

H. O. Fattorini and D. L. Russell, Uniform bounds on biorthogonal functions for real exponentials with an application to control theory of parabolic equations,, Quarterly of applied Maths, 32 (1974), 45. Google Scholar

[21]

E. Fernandez-Cara, Null controllability of the semilinear heat equation,, ESAIM Control Optim. Calc. Var., 2 (1997), 87. doi: 10.1051/cocv:1997104. Google Scholar

[22]

E. Fernandez-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations,, Annales de l'Institut Henry Poincaré, 17 (2000), 583. doi: 10.1016/S0294-1449(00)00117-7. Google Scholar

[23]

A. V. Fursikov and O. Y. Imanuvilov, "Controllability of Evolution Equations,", Lectures Notes Series 34, (1996). Google Scholar

[24]

M. Gonzalez-Burgos and L. de Teresa, Controllability results for cascade systems of m coupled parabolic PDEs by one control force,, Portugal. Math., 67 (2010), 91. doi: 10.4171/PM/1859. Google Scholar

[25]

M. Gonzalez-Burgos and R. Perez-Garcia, Controllability of some coupled parabolic systems by one control force,, C. R. Math. Acad. Sci. Paris, 340 (2005), 125. doi: 10.1016/j.crma.2004.11.025. Google Scholar

[26]

M. Gonzalez-Burgos and R. Perez-Garcia, Controllability results for some nonlinear coupled parabolic systems by one control force,, Asymptot. Anal., 46 (2006), 123. Google Scholar

[27]

S. Guerrero, Null controllability of some systems of two parabolic equations with one control force,, SIAM J. Control Optim., 46 (2007), 379. doi: 10.1137/060653135. Google Scholar

[28]

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

[29]

X. Liu, H. Gao and P. Lin, Null controllability of a cascade system of degenerate parabolic equations,, Acta Math. Sci. Ser. A Chin. Ed., 28 (2008), 985. Google Scholar

[30]

A. Lopez, X. Zhang and E. Zuazua, Null controllability of the heat equation as singular limit of the exact controllability of dissipative wave equations,, J. Math. Pures Appl., 79 (2000), 741. doi: 10.1016/S0021-7824(99)00144-0. Google Scholar

[31]

P. Martinez and J. Vancostenoble, Carleman estimates for one-dimensional degenerate heat equations,, J. Evol. Equ., 6 (2006), 325. doi: 10.1007/s00028-006-0214-6. Google Scholar

[32]

S. Micu and E. Zuazua, On the lack of null controllability of the heat equation on the half-line,, Trans. Amer. Math. Soc., 353 (2001), 1635. doi: 10.1090/S0002-9947-00-02665-9. Google Scholar

[33]

B. Opic and A. Kufner, "Hardy-Type Inequalities,", Longman Scientific and Technical, (1990). Google Scholar

[34]

D. L. Russell, A unified boundary controllability theory for hyperbolic and parabolic partial differential equations,, Studies in Applied Mathematics, 52 (1973), 189. Google Scholar

[35]

D. Tataru, A priori estimates of Carleman's type in domains with boundary,, J. Math. Pures et Appliquées, 73 (1994), 355. Google Scholar

[36]

E. Zuazua, Approximate controllability for semilinear heat equations with globally Lipschitz nonlinearites,, Control Cybernet., 28 (1999), 665. Google Scholar

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