# American Institute of Mathematical Sciences

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

## Controllability of fast diffusion coupled parabolic systems

 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.
Citation: Felipe Wallison Chaves-Silva, Sergio Guerrero, Jean Pierre Puel. Controllability of fast diffusion coupled parabolic systems. Mathematical Control and 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), 928-943. doi: 10.1016/j.jmaa.2005.07.060. [2] M. Bendahmane and F. W. Chaves-Silva, Uniform null controllability for a degenerating reaction-diffusion system approximating a simplified cardiac model,, preprint, (). [3] J.-M. Coron and S. Guerrero, A singular optimal control: A linear 1-D parabolic hyperbolic example, Asymptot. Analisys, 44 (2005), 237-257. [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), 285-308. doi: 10.1007/s00574-013-0014-x. [5] A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series 34, Research Institute of Mathematics, Seoul National University, Seoul, 1996. [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), 852-868. doi: 10.1016/j.jfa.2009.06.035. [7] M. González-Burgos and R. Pérez-García, Controllability results for some nonlinear coupled parabolic systems by one control force, Asymptot. Anal., 46 (2006), 123-162. [8] S. Guerrero, Null controllability of some systems of two parabolic equations with one control force, SIAM J. Control Optim., 46 (2007), 379-394. doi: 10.1137/060653135. [9] S. Guerrero and G. Lebeau, Singular optimal control for a transport-diffusion equation, Comm. Partial Differential Equations, 32 (2007), 1813-1836. doi: 10.1080/03605300701743756. [10] D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, I, Jahresber. Dtsch. Math.-Ver, 105 (2003), 103-165. [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), 741-808. doi: 10.1016/S0021-7824(99)00144-0. [12] J.-L. Lions, Some Methods in Mathematical Analysis of System and their Control, Science Press, Beijing, China, Gordon and Breach, New York, 1981. [13] J.-L. Lions and E. Magenes, Problèmes aux Limites Non Homogènes et Applications, volumes 1, 2 et 3, Dunod, Paris, 1968.

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), 928-943. doi: 10.1016/j.jmaa.2005.07.060. [2] M. Bendahmane and F. W. Chaves-Silva, Uniform null controllability for a degenerating reaction-diffusion system approximating a simplified cardiac model,, preprint, (). [3] J.-M. Coron and S. Guerrero, A singular optimal control: A linear 1-D parabolic hyperbolic example, Asymptot. Analisys, 44 (2005), 237-257. [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), 285-308. doi: 10.1007/s00574-013-0014-x. [5] A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series 34, Research Institute of Mathematics, Seoul National University, Seoul, 1996. [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), 852-868. doi: 10.1016/j.jfa.2009.06.035. [7] M. González-Burgos and R. Pérez-García, Controllability results for some nonlinear coupled parabolic systems by one control force, Asymptot. Anal., 46 (2006), 123-162. [8] S. Guerrero, Null controllability of some systems of two parabolic equations with one control force, SIAM J. Control Optim., 46 (2007), 379-394. doi: 10.1137/060653135. [9] S. Guerrero and G. Lebeau, Singular optimal control for a transport-diffusion equation, Comm. Partial Differential Equations, 32 (2007), 1813-1836. doi: 10.1080/03605300701743756. [10] D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, I, Jahresber. Dtsch. Math.-Ver, 105 (2003), 103-165. [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), 741-808. doi: 10.1016/S0021-7824(99)00144-0. [12] J.-L. Lions, Some Methods in Mathematical Analysis of System and their Control, Science Press, Beijing, China, Gordon and Breach, New York, 1981. [13] J.-L. Lions and E. Magenes, Problèmes aux Limites Non Homogènes et Applications, volumes 1, 2 et 3, Dunod, Paris, 1968.
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