Exact controllability to the trajectories of the heat equation with Fourier boundary conditions: the semilinear case

Enrique Fernández-Cara; Manuel González-Burgos; Sergio Guerrero; Jean-Pierre Puel

ESAIM: Control, Optimisation and Calculus of Variations (2006)

  • Volume: 12, Issue: 3, page 466-483
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Fernández-Cara, Enrique, et al. "Exact controllability to the trajectories of the heat equation with Fourier boundary conditions: the semilinear case." ESAIM: Control, Optimisation and Calculus of Variations 12.3 (2006): 466-483. <http://eudml.org/doc/249672>.

@article{Fernández2006,
abstract = { 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 $\{\partial y\over\partial n\} + f(y) = 0$. 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. },
author = {Fernández-Cara, Enrique, González-Burgos, Manuel, Guerrero, Sergio, Puel, Jean-Pierre},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Controllability; heat equation; Fourier boundary conditions; semilinear.; controllability; semilinear},
language = {eng},
month = {6},
number = {3},
pages = {466-483},
publisher = {EDP Sciences},
title = {Exact controllability to the trajectories of the heat equation with Fourier boundary conditions: the semilinear case},
url = {http://eudml.org/doc/249672},
volume = {12},
year = {2006},
}

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
DA - 2006/6//
PB - EDP Sciences
VL - 12
IS - 3
SP - 466
EP - 483
AB - 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 ${\partial y\over\partial n} + f(y) = 0$. 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.
LA - eng
KW - Controllability; heat equation; Fourier boundary conditions; semilinear.; controllability; semilinear
UR - http://eudml.org/doc/249672
ER -

References

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  11. E. Fernández-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations. Ann. Inst. H. Poincaré, Anal. non Linéaire17 (2000) 583–616.  Zbl0970.93023
  12. A. Fursikov and O.Yu. Imanuvilov, Controllability of Evolution Equations. Lecture Notes #34, Seoul National University, Korea (1996).  
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  14. I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. Cambridge University Press, Cambridge (2000).  Zbl0942.93001
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