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December  2014, 4(4): 465-479. doi: 10.3934/mcrf.2014.4.465

Controllability of fast diffusion coupled parabolic systems

Felipe Wallison Chaves-Silva 1, , Sergio Guerrero 2, and  Jean Pierre Puel 3,

1. 

BCAM - Basque Center for Applied Mathematics, Mazarredo 14, E-48009 Bilbao, Basque Country, Spain
2. 

Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie - Paris 6, Boîte Corrier 187, F-75252, Paris Cedex 05, France
3. 

Laboratoire de Mathématiques de Versailles, Université de Versailles - St. Quentin, 45 Avenue des Etats Unis, 78035 Versailles

Received  March 2013 Revised  October 2013 Published  September 2014

In this work we are concerned with the null controllability of coupled parabolic systems depending on a parameter and converging to a parabolic-elliptic system. We show the uniform null controllability of the family of coupled parabolic systems with respect to the degenerating parameter.
Keywords: null controllability, Fast diffusion parabolic equations, Carleman estimates..
Mathematics Subject Classification: Primary: 35K65, 93B05; Secondary: 93B0.
Citation: 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
References:
[1]

F. Ammar-Khodja, A. Benabdallah and C. Dupaix, Null controllability of 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

[2]

M. Bendahmane and F. W. Chaves-Silva, Uniform null controllability for a degenerating reaction-diffusion system approximating a simplified cardiac model,, preprint, ().   Google Scholar

[3]

J.-M. Coron and S. Guerrero, A singular optimal control: A linear 1-D parabolic hyperbolic example,, Asymptot. Analisys, 44 (2005), 237.   Google Scholar

[4]

E. Fernandéz-Cara, J. Limaco and S. B. de Menezes, Null controllability for a parabolic-elliptic coupled system,, Bull. Braz. Math. Soc. (N.S.), 44 (2013), 285.  doi: 10.1007/s00574-013-0014-x.  Google Scholar

[5]

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

[6]

O. Glass, A complex-analytic approach to the problem of uniform controllability of a transport equation in the vanishing viscosity limit,, J. Funct. Analysis, 258 (2010), 852.  doi: 10.1016/j.jfa.2009.06.035.  Google Scholar

[7]

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

[8]

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

[9]

S. Guerrero and G. Lebeau, Singular optimal control for a transport-diffusion equation,, Comm. Partial Differential Equations, 32 (2007), 1813.  doi: 10.1080/03605300701743756.  Google Scholar

[10]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, I,, Jahresber. Dtsch. Math.-Ver, 105 (2003), 103.   Google Scholar

[11]

A. Lopes, 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

[12]

J.-L. Lions, Some Methods in Mathematical Analysis of System and their Control,, Science Press, (1981).   Google Scholar

[13]

J.-L. Lions and E. Magenes, Problèmes aux Limites Non Homogènes et Applications,, volumes 1, (1968).   Google Scholar

show all references

References:
[1]

F. Ammar-Khodja, A. Benabdallah and C. Dupaix, Null controllability of 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

[2]

M. Bendahmane and F. W. Chaves-Silva, Uniform null controllability for a degenerating reaction-diffusion system approximating a simplified cardiac model,, preprint, ().   Google Scholar

[3]

J.-M. Coron and S. Guerrero, A singular optimal control: A linear 1-D parabolic hyperbolic example,, Asymptot. Analisys, 44 (2005), 237.   Google Scholar

[4]

E. Fernandéz-Cara, J. Limaco and S. B. de Menezes, Null controllability for a parabolic-elliptic coupled system,, Bull. Braz. Math. Soc. (N.S.), 44 (2013), 285.  doi: 10.1007/s00574-013-0014-x.  Google Scholar

[5]

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

[6]

O. Glass, A complex-analytic approach to the problem of uniform controllability of a transport equation in the vanishing viscosity limit,, J. Funct. Analysis, 258 (2010), 852.  doi: 10.1016/j.jfa.2009.06.035.  Google Scholar

[7]

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

[8]

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

[9]

S. Guerrero and G. Lebeau, Singular optimal control for a transport-diffusion equation,, Comm. Partial Differential Equations, 32 (2007), 1813.  doi: 10.1080/03605300701743756.  Google Scholar

[10]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, I,, Jahresber. Dtsch. Math.-Ver, 105 (2003), 103.   Google Scholar

[11]

A. Lopes, 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

[12]

J.-L. Lions, Some Methods in Mathematical Analysis of System and their Control,, Science Press, (1981).   Google Scholar

[13]

J.-L. Lions and E. Magenes, Problèmes aux Limites Non Homogènes et Applications,, volumes 1, (1968).   Google Scholar

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