Geodesic
Exact controllability to the trajectories of the heat equation with Fourier boundary conditions : the semilinear 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. 466-483.

Voir la notice de l'article provenant de la source Numdam

This paper is concerned with the global exact controllability of the semilinear heat equation (with nonlinear terms involving the state and the gradient) completed with boundary conditions of the form ∂y ∂n+f(y)=0. We consider distributed controls, with support in a small set. The null controllability of similar linear systems has been analyzed in a previous first part of this work. In this second part we show that, when the nonlinear terms are locally Lipschitz-continuous and slightly superlinear, one has exact controllability to the trajectories.

DOI : 10.1051/cocv:2006011
Classification : 35K20, 93B05
Keywords: controllability, heat equation, Fourier boundary conditions, semilinear

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_466_0,
     author = {Fern\'andez-Cara, Enrique and Gonz\'alez-Burgos, Manuel and Guerrero, Sergio and Puel, Jean-Pierre},
     title = {Exact controllability to the trajectories of the heat equation with {Fourier} boundary conditions : the semilinear case},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {466--483},
     publisher = {EDP-Sciences},
     volume = {12},
     number = {3},
     year = {2006},
     doi = {10.1051/cocv:2006011},
     mrnumber = {2224823},
     zbl = {1106.93010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv:2006011/}
}
TY  - JOUR
AU  - Fernández-Cara, Enrique
AU  - González-Burgos, Manuel
AU  - Guerrero, Sergio
AU  - Puel, Jean-Pierre
TI  - Exact controllability to the trajectories of the heat equation with Fourier boundary conditions : the semilinear case
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2006
SP  - 466
EP  - 483
VL  - 12
IS  - 3
PB  - EDP-Sciences
UR  - http://geodesic.mathdoc.fr/articles/10.1051/cocv:2006011/
DO  - 10.1051/cocv:2006011
LA  - en
ID  - COCV_2006__12_3_466_0
ER  - 
%0 Journal Article
%A Fernández-Cara, Enrique
%A González-Burgos, Manuel
%A Guerrero, Sergio
%A Puel, Jean-Pierre
%T Exact controllability to the trajectories of the heat equation with Fourier boundary conditions : the semilinear case
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2006
%P 466-483
%V 12
%N 3
%I EDP-Sciences
%U http://geodesic.mathdoc.fr/articles/10.1051/cocv:2006011/
%R 10.1051/cocv:2006011
%G en
%F COCV_2006__12_3_466_0
Fernández-Cara, Enrique; González-Burgos, Manuel; Guerrero, Sergio; Puel, Jean-Pierre. Exact controllability to the trajectories of the heat equation with Fourier boundary conditions : the semilinear case. ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 3, pp. 466-483. doi : 10.1051/cocv:2006011. http://geodesic.mathdoc.fr/articles/10.1051/cocv:2006011/

[1] H. Amann, Parabolic evolution equations and nonlinear boundary conditions. J. Diff. Equ. 72 (1988) 201-269. | Zbl

[2] J. Arrieta, A. Carvalho and A. Rodríguez-Bernal, Parabolic problems with nonlinear boundary conditions and critical nonlinearities. J. Diff. Equ. 156 (1999) 376-406. | Zbl

[3] J.P. Aubin, L'analyse non linéaire et ses motivations économiques. Masson, Paris (1984). | Zbl

[4] O. Bodart, M. González-Burgos and R. Pŕez-García, Insensitizing controls for a semilinear heat equation with a superlinear nonlinearity. C. R. Math. Acad. Sci. Paris 335 (2002) 677-682. | Zbl

[5] 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

[6] A. Doubova, E. Fernández-Cara, M. González-Burgos and E. Zuazua, On the controllability of parabolic systems with a nonlinear term involving the state and the gradient. SIAM J. Control Optim. 41 (2002) 798-819. | Zbl

[7] L. Evans, Regularity properties of the heat equation subject to nonlinear boundary constraints. Nonlinear Anal. 1 (1997) 593-602. | Zbl

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

[9] L.A. Fernández and E. Zuazua, Approximate controllability for the semi-linear heat equation involving gradient terms. J. Optim. Theory Appl. 101 (1999) 307-328. | Zbl

[10] E. Fernández-Cara, M. González-Burgos, S. Guerrero and J.P. Puel, Null controllability of the heat equation with boundary Fourier conditions: The linear case. ESAIM: COCV 12 442-465. | Zbl | mathdoc-id

[11] E. Fernández-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations. Ann. Inst. H. Poincaré, Anal. non Linéaire 17 (2000) 583-616. | Zbl | mathdoc-id

[12] A. Fursikov and O.Yu. Imanuvilov, Controllability of Evolution Equations. Lecture Notes #34, Seoul National University, Korea (1996). | Zbl | MR

[13] I. Lasiecka and R. Triggiani, Exact controllability of semilinear abstract systems with applications to waves and plates boundary control. Appl. Math. Optim. 23 (1991) 109-154. | Zbl

[14] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. Cambridge University Press, Cambridge (2000). | Zbl

[15] E. Zuazua, Exact boundary controllability for the semilinear wave equation, in Nonlinear Partial Differential Equations and their Applications, Vol. X, H. Brezis and J.L. Lions Eds. Pitman (1991) 357-391. | Zbl

[16] E. Zuazua, Exact controllability for the semilinear wave equation in one space dimension. Ann. I.H.P., Analyse non Linéaire 10 (1993) 109-129. | Zbl | mathdoc-id

Cité par Sources :