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Exact controllability to trajectories for semilinear heatequations with discontinuous diffusion coefficients

Published online by Cambridge University Press:  15 August 2002

Anna Doubova ,
A. Osses  and
J.-P. Puel
Anna Doubova
Affiliation:
Departamento E.D.A.N., Universidad de Sevilla, Tarfia s/n, 41012 Sevilla, Spain and École Polytechnique, 91128 Palaiseau Cedex, France; [email protected]. [email protected]. This work has been partially supported by D.G.E.S., Spain, Grants PB98–1134.
A. Osses
Affiliation:
Departamento de Ingenería Matemática, Facultad de Ciencias de Físicas y Matemáticas, Universidad de Chile, Casilla 170/3 - Correo 3, Santiago, Chile and Centro de Modelamiento Matemático, UMR 2071 CNRS-Uchile; [email protected]. This work has been partially supported by FONDECYT grants No. 1000955 and 7000955.
J.-P. Puel
Affiliation:
Laboratoire de Mathématiques Appliquées, Université de Versailles Saint-Quentin, 45 avenue des États Unis, 78035 Versailles Cedex, France and École Polytechnique, 91128 Palaiseau Cedex, France; [email protected].
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Abstract

The results of this paper concern exact controllability to thetrajectories for a coupled system of semilinear heat equations. Wehave transmission conditions on the interface and Dirichlet boundaryconditions at the external part of the boundary so that the system can beviewed as a single equation with discontinuous coefficients in theprincipal part. Exact controllability to the trajectories is proved when weconsider distributed controls supported in the part of the domain where thediffusion coefficient is the smaller and if the nonlinear term f(y) growsslower than |y|log3/2(1+|y|) at infinity. In the proof we use nullcontrollability results for the associate linear system and globalCarleman estimates with explicit bounds or combinations of several ofthese estimates. In order to treat the terms appearing on theinterface, we have to construct specific weight functions depending ongeometry.

Keywords

Carleman inequalitiescontrollabilitytransmission problems.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 8: A tribute to JL Lions , 2002 , pp. 621 - 661
Copyright
© EDP Sciences, SMAI, 2002

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