no. 3
Null controllability of the heat equation with boundary Fourier conditions : the linear case
Fernández-Cara, Enrique ; González-Burgos, Manuel ; Guerrero, Sergio1 ; Puel, Jean-Pierre2
1 Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, boîte courrier 187, 75035 Cedex 05, Paris, France;
2 Laboratoire de Mathématiques Appliquées, Université de Versailles – St. Quentin, 45 avenue des États-Unis, 78035 Versailles, France; ; Laboratoire de Mathématiques Appliquées, Université de Versailles, St. Quentin, 45 avenue des États-Unis, 78035 Versailles, France;
ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 3, pp. 442-465.

In this paper, we prove the global null controllability of the linear heat equation completed with linear Fourier boundary conditions of the form ∂y ∂n+βy=0. We consider distributed controls with support in a small set and nonregular coefficients β=β(x,t). For the proof of null controllability, a crucial tool will be a new Carleman estimate for the weak solutions of the classical heat equation with nonhomogeneous Neumann boundary conditions.

DOI : 10.1051/cocv:2006010
Classification : 35K20, 93B05
Mots clés : controllability, heat equation, Fourier conditions
Fernández-Cara, Enrique  ; González-Burgos, Manuel  ; Guerrero, Sergio 1 ; Puel, Jean-Pierre 2

1 Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, boîte courrier 187, 75035 Cedex 05, Paris, France;
2 Laboratoire de Mathématiques Appliquées, Université de Versailles – St. Quentin, 45 avenue des États-Unis, 78035 Versailles, France; ; Laboratoire de Mathématiques Appliquées, Université de Versailles, St. Quentin, 45 avenue des États-Unis, 78035 Versailles, France; 
@article{COCV_2006__12_3_442_0,
     author = {Fern\'andez-Cara, Enrique and Gonz\'alez-Burgos, Manuel and Guerrero, Sergio and Puel, Jean-Pierre},
     title = {Null controllability of the heat equation with boundary {Fourier} conditions : the linear case},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {442--465},
     publisher = {EDP-Sciences},
     volume = {12},
     number = {3},
     year = {2006},
     doi = {10.1051/cocv:2006010},
     mrnumber = {2224822},
     zbl = {1106.93009},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv:2006010/}
}
TY  - JOUR
AU  - Fernández-Cara, Enrique
AU  - González-Burgos, Manuel
AU  - Guerrero, Sergio
AU  - Puel, Jean-Pierre
TI  - Null controllability of the heat equation with boundary Fourier conditions : the linear case
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2006
SP  - 442
EP  - 465
VL  - 12
IS  - 3
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv:2006010/
DO  - 10.1051/cocv:2006010
LA  - en
ID  - COCV_2006__12_3_442_0
ER  - 
%0 Journal Article
%A Fernández-Cara, Enrique
%A González-Burgos, Manuel
%A Guerrero, Sergio
%A Puel, Jean-Pierre
%T Null controllability of the heat equation with boundary Fourier conditions : the linear case
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2006
%P 442-465
%V 12
%N 3
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv:2006010/
%R 10.1051/cocv:2006010
%G en
%F COCV_2006__12_3_442_0
Fernández-Cara, Enrique; González-Burgos, Manuel; Guerrero, Sergio; Puel, Jean-Pierre. Null controllability of the heat equation with boundary Fourier conditions : the linear case. ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 3, pp. 442-465. doi : 10.1051/cocv:2006010. http://www.numdam.org/articles/10.1051/cocv:2006010/

[1] V. Barbu, Controllability of parabolic and Navier-Stokes equations. Sci. Math. Jpn 56 (2002) 143-211. | Zbl

[2] A. Doubova, E. Fernández-Cara and M. González-Burgos, On the controllability of the heat equation with nonlinear boundary Fourier conditions. J. Diff. Equ. 196 (2004) 385-417. | Zbl

[3] C. Fabre, J.P. Puel and E. Zuazua, Approximate controllability of the semilinear heat equation. Proc. Roy. Soc. Edinburgh 125A (1995) 31-61. | Zbl

[4] E. Fernández-Cara and E. Zuazua, The cost of approximate controllability for heat equations: the linear case. Adv. Diff. Equ. 5 (2000) 465-514. | Zbl

[5] A. Fursikov and O.Yu. Imanuvilov, Controllability of Evolution Equations. Lecture Notes no. 34, Seoul National University, Korea, 1996. | MR | Zbl

[6] O.Yu. Imanuvilov and M. Yamamoto, Carleman estimate for a parabolic equation in a Sobolev space of negative order and its applications, Dekker, New York. Lect. Notes Pure Appl. Math. 218 (2001). | MR | Zbl

[7] G. Lebeau and L. Robbiano, Contrôle exacte de l'equation de la chaleur (French). Comm. Partial Differ. Equat. 20 (1995) 335-356. | Zbl

[8] D.L. Russell, A unified boundary controllability theory for hyperbolic and parabolic partial differential equations. Studies Appl. Math. 52 (1973) 189-211. | Zbl

Cité par Sources :