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Null controllability of the heat equation withboundary Fourier conditions: the linear case

Published online by Cambridge University Press:  20 June 2006

Enrique Fernández-Cara ,
Manuel González-Burgos ,
Sergio Guerrero  and
Jean-Pierre Puel
Enrique Fernández-Cara
Affiliation:
Dpto. E.D.A.N., Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain; [email protected]; [email protected]; [email protected]
Manuel González-Burgos
Affiliation:
Dpto. E.D.A.N., Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain; [email protected]; [email protected]; [email protected]
Sergio Guerrero
Affiliation:
Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, boîte courrier 187, 75035 Cedex 05, Paris, France; [email protected]
Jean-Pierre Puel
Affiliation:
Laboratoire de Mathématiques Appliquées, Université de Versailles, St. Quentin, 45 avenue des États-Unis, 78035 Versailles, France; [email protected]
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Abstract

In this paper, we prove the global null controllability ofthe linear heat equation completed with linear Fourierboundary conditions of the form ${\partial y\over\partial n} + \beta\,y = 0$ . We consider distributed controls with support in a small set andnonregular coefficients $\beta=\beta(x,t)$ . For the proof of null controllability, a crucial tool will be a new Carleman estimate for the weak solutions of the classical heatequation with nonhomogeneous Neumann boundary conditions.

Keywords

Controllabilityheat equationFourier conditions.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 12 , Issue 3 , July 2006 , pp. 442 - 465
Copyright
© EDP Sciences, SMAI, 2006

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References

Barbu, V., Controllability of parabolic and Navier-Stokes equations. Sci. Math. Jpn 56 (2002) 143–211.
Doubova, A., Fernández-Cara, E. and González-Burgos, M., On the controllability of the heat equation with nonlinear boundary Fourier conditions. J. Diff. Equ. 196 (2004) 385–417. CrossRef
Fabre, C., Puel, J.P. and Zuazua, E., Approximate controllability of the semilinear heat equation. Proc. Roy. Soc. Edinburgh 125A (1995) 31–61. CrossRef
Fernández-Cara, E. and Zuazua, E., The cost of approximate controllability for heat equations: the linear case. Adv. Diff. Equ. 5 (2000) 465–514.
A. Fursikov and O.Yu. Imanuvilov, Controllability of Evolution Equations. Lecture Notes no. 34, Seoul National University, Korea, 1996.
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).
Lebeau, G. and Robbiano, L., Contrôle exacte de l'equation de la chaleur (French). Comm. Partial Differ. Equat. 20 (1995) 335–356. CrossRef
Russell, D.L., A unified boundary controllability theory for hyperbolic and parabolic partial differential equations. Studies Appl. Math. 52 (1973) 189–211. CrossRef