March  2014, 4(1): 1-15. doi: 10.3934/mcrf.2014.4.1

Null controllability of retarded parabolic equations

1. 

Laboratoire de Mathématiques de Besançon, UMR CNRS 6623, Université de Franche-Comté, F25030 Besançon Cedex, France, France

2. 

École Nationale Supérieure des Travaux Publics, Rue Sidi Garidi, BP 32, 16051 Alger, Algeria

3. 

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

Received  May 2012 Revised  December 2012 Published  December 2013

We address in this work the null controllability problem for a linear heat equation with delay parameters. The control is exerted on a subdomain and we show how the global Carleman estimate due to Fursikov and Imanuvilov can be applied to derive results in this direction.
Citation: 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
References:
[1]

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.  doi: 10.1007/s00028-009-0008-8.  Google Scholar

[2]

F. Ammar-Khodja, A. Benabdallah, C. Dupaix and M. González-Burgos, Controllability for a class of reaction-diffusion systems: The generalized Kalman's condition,, C. R. Math. Acad. Sci. Paris, 345 (2007), 543.  doi: 10.1016/j.crma.2007.10.023.  Google Scholar

[3]

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,, Differ. Equ. Appl., 1 (2009), 427.  doi: 10.7153/dea-01-24.  Google Scholar

[4]

F. Ammar-Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa, The Kalman condition for the boundary controllability of coupled parabolic systems. Bounds on biorthogonal families to complex matrix exponentials,, J. Math. Pures Appl. (9), 96 (2011), 555.  doi: 10.1016/j.matpur.2011.06.005.  Google Scholar

[5]

M. Artola, Sur les perturbations des équations d'évolution: Application à des problèmes avec retard,, Ann. Sci. École Norm. Sup. (4), 2 (1969), 137.   Google Scholar

[6]

V. Barbu, Exact controllability of the superlinear heat equation,, Appl. Math. Optim., 42 (2000), 73.  doi: 10.1007/s002450010004.  Google Scholar

[7]

R. F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory,, Texts in Applied Mathematics, (1995).  doi: 10.1007/978-1-4612-4224-6.  Google Scholar

[8]

E. Fernández-Cara, M. González-Burgos and L. de Teresa, Boundary controllability of parabolic coupled equations,, J. Funct. Anal., 259 (2010), 1720.  doi: 10.1016/j.jfa.2010.06.003.  Google Scholar

[9]

E. Fernández-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations,, Ann. Inst. H. Poincaré, 17 (2000), 583.  doi: 10.1016/S0294-1449(00)00117-7.  Google Scholar

[10]

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

[11]

S.-I. Nakagiri, Optimal control of linear retarded systems in Banach spaces,, J. Math. Anal. and Appl., 120 (1986), 169.  doi: 10.1016/0022-247X(86)90210-6.  Google Scholar

[12]

S.-I. Nakagiri and M. Yamamoto, Controllability and observability of linear retarded systems in Banach spaces,, Int. J. Control, 49 (1989), 1489.   Google Scholar

[13]

J. Zabczyk, Mathematical Control Theory: An Introduction,, Systems & Control: Foundations & Applications, (1992).   Google Scholar

show all references

References:
[1]

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.  doi: 10.1007/s00028-009-0008-8.  Google Scholar

[2]

F. Ammar-Khodja, A. Benabdallah, C. Dupaix and M. González-Burgos, Controllability for a class of reaction-diffusion systems: The generalized Kalman's condition,, C. R. Math. Acad. Sci. Paris, 345 (2007), 543.  doi: 10.1016/j.crma.2007.10.023.  Google Scholar

[3]

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,, Differ. Equ. Appl., 1 (2009), 427.  doi: 10.7153/dea-01-24.  Google Scholar

[4]

F. Ammar-Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa, The Kalman condition for the boundary controllability of coupled parabolic systems. Bounds on biorthogonal families to complex matrix exponentials,, J. Math. Pures Appl. (9), 96 (2011), 555.  doi: 10.1016/j.matpur.2011.06.005.  Google Scholar

[5]

M. Artola, Sur les perturbations des équations d'évolution: Application à des problèmes avec retard,, Ann. Sci. École Norm. Sup. (4), 2 (1969), 137.   Google Scholar

[6]

V. Barbu, Exact controllability of the superlinear heat equation,, Appl. Math. Optim., 42 (2000), 73.  doi: 10.1007/s002450010004.  Google Scholar

[7]

R. F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory,, Texts in Applied Mathematics, (1995).  doi: 10.1007/978-1-4612-4224-6.  Google Scholar

[8]

E. Fernández-Cara, M. González-Burgos and L. de Teresa, Boundary controllability of parabolic coupled equations,, J. Funct. Anal., 259 (2010), 1720.  doi: 10.1016/j.jfa.2010.06.003.  Google Scholar

[9]

E. Fernández-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations,, Ann. Inst. H. Poincaré, 17 (2000), 583.  doi: 10.1016/S0294-1449(00)00117-7.  Google Scholar

[10]

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

[11]

S.-I. Nakagiri, Optimal control of linear retarded systems in Banach spaces,, J. Math. Anal. and Appl., 120 (1986), 169.  doi: 10.1016/0022-247X(86)90210-6.  Google Scholar

[12]

S.-I. Nakagiri and M. Yamamoto, Controllability and observability of linear retarded systems in Banach spaces,, Int. J. Control, 49 (1989), 1489.   Google Scholar

[13]

J. Zabczyk, Mathematical Control Theory: An Introduction,, Systems & Control: Foundations & Applications, (1992).   Google Scholar

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