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Approximate controllability of the semilinear heat equation

Published online by Cambridge University Press:  14 November 2011

Caroline Fabre ,
Jean-Pierre Puel  and
Enrike Zuazua
Caroline Fabre
Affiliation:
Université Paris XII-Val de Marne, U.F.R. Sciences, Laboratoire de Mathématiques, Av. du Général de Gaulle, 94010 Creteil Cedex and Centre de Mathématiques Appliquées, Ecole Polytechnique, 91128 Palaiseau Cedex, France
Jean-Pierre Puel
Affiliation:
Université de Versailles Saint-Quentin, Département de Mathématiques and Centre de Mathématiques Appliquées, Ecole Polytechnique, 91128 Palaiseau Cedex, France
Enrike Zuazua
Affiliation:
Departamento de Matemática Aplicada, Universidad Complutense, 28040 Madrid, Spain
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Abstract

This article is concerned with the study of approximate controllability for the semilinear heat equation in a bounded domain Ω when the control acts on any open and nonempty subset of Ω or on a part of the boundary. In the case of both an internal and a boundary control, the approximate controllability in LP(Ω) for 1 ≦ p < + ∞ is proved when the nonlinearity is globally Lipschitz with a control in L∞. In the case of the interior control, we also prove approximate controllability in C0(Ω). The proof combines a variational approach to the controllability problem for linear equations and a fixed point method. We also prove that the control can be taken to be of “quasi bang-bang” form.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 125 , Issue 1 , 1995 , pp. 31 - 61
Copyright
Copyright © Royal Society of Edinburgh 1995

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