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 . 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.
Classification : 35K20, 93B05
Mots clés : controllability, heat equation, Fourier boundary conditions, semilinear
@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},
zbl = {1106.93010},
mrnumber = {2224823},
language = {en},
url = {http://www.numdam.org/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 DA - 2006/// SP - 466 EP - 483 VL - 12 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2006011/ UR - https://zbmath.org/?q=an%3A1106.93010 UR - https://www.ams.org/mathscinet-getitem?mr=2224823 UR - https://doi.org/10.1051/cocv:2006011 DO - 10.1051/cocv:2006011 LA - en ID - COCV_2006__12_3_466_0 ER -
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://www.numdam.org/articles/10.1051/cocv:2006011/
[1] , Parabolic evolution equations and nonlinear boundary conditions. J. Diff. Equ. 72 (1988) 201-269. | Zbl 0658.34011
[2] , and, Parabolic problems with nonlinear boundary conditions and critical nonlinearities. J. Diff. Equ. 156 (1999) 376-406. | Zbl 0938.35077
[3] , L'analyse non linéaire et ses motivations économiques. Masson, Paris (1984). | Zbl 0551.90001
[4] , and, Insensitizing controls for a semilinear heat equation with a superlinear nonlinearity. C. R. Math. Acad. Sci. Paris 335 (2002) 677-682. | Zbl 1021.35049
[5] , and, On the controllability of the heat equation with nonlinear boundary Fourier conditions. J. Diff. Equ. 196 (2004) 385-417. | Zbl 1049.35042
[6] ,, and, 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 1038.93041
[7] , Regularity properties of the heat equation subject to nonlinear boundary constraints. Nonlinear Anal. 1 (1997) 593-602. | Zbl 0369.35034
[8] , and, Approximate controllability of the semilinear heat equation. Proc. Roy. Soc. Edinburgh 125A (1995) 31-61. | Zbl 0818.93032
[9] and, Approximate controllability for the semi-linear heat equation involving gradient terms. J. Optim. Theory Appl. 101 (1999) 307-328. | Zbl 0952.49003
[10] ,, and, Null controllability of the heat equation with boundary Fourier conditions: The linear case. ESAIM: COCV 12 442-465. | Numdam | Zbl 1106.93009
[11] and, Null and approximate controllability for weakly blowing up semilinear heat equations. Ann. Inst. H. Poincaré, Anal. non Linéaire 17 (2000) 583-616. | Numdam | Zbl 0970.93023
[12] and, Controllability of Evolution Equations. Lecture Notes #34, Seoul National University, Korea (1996). | MR 1406566 | Zbl 0862.49004
[13] and, Exact controllability of semilinear abstract systems with applications to waves and plates boundary control. Appl. Math. Optim. 23 (1991) 109-154. | Zbl 0729.93023
[14] and, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. Cambridge University Press, Cambridge (2000). | Zbl 0961.93003
[15] , 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 0731.93011
[16] , Exact controllability for the semilinear wave equation in one space dimension. Ann. I.H.P., Analyse non Linéaire 10 (1993) 109-129. | Numdam | Zbl 0769.93017
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