Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

Approximate controllability of the semilinear heat equation

Caroline Fabrea1, Jean-Pierre Puela2 and Enrike Zuazuaa3

a1 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

a2 Université de Versailles Saint-Quentin, Département de Mathématiques and Centre de Mathématiques Appliquées, Ecole Polytechnique, 91128 Palaiseau Cedex, France

a3 Departamento de Matemática Aplicada, Universidad Complutense, 28040 Madrid, Spain

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.

(Received January 25 1993)

(Revised May 13 1993)