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.

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 : https://doi.org/10.1051/cocv:2006011
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/}
}
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] H. Amann, Parabolic evolution equations and nonlinear boundary conditions. J. Diff. Equ. 72 (1988) 201-269. | Zbl 0658.34011

[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 0938.35077

[3] J.P. Aubin, L'analyse non linéaire et ses motivations économiques. Masson, Paris (1984). | Zbl 0551.90001

[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 1021.35049

[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 1049.35042

[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 1038.93041

[7] L. Evans, Regularity properties of the heat equation subject to nonlinear boundary constraints. Nonlinear Anal. 1 (1997) 593-602. | Zbl 0369.35034

[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 0818.93032

[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 0952.49003

[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. | Numdam | Zbl 1106.93009

[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. | Numdam | Zbl 0970.93023

[12] A. Fursikov and O.Yu. Imanuvilov, Controllability of Evolution Equations. Lecture Notes #34, Seoul National University, Korea (1996). | MR 1406566 | Zbl 0862.49004

[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 0729.93023

[14] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. Cambridge University Press, Cambridge (2000). | Zbl 0961.93003

[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 0731.93011

[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. | Numdam | Zbl 0769.93017