Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients

¹ Departamento E.D.A.N., Universidad de Sevilla, Tarfia s/n, 41012 Sevilla, Spain and École Polytechnique, 91128 Palaiseau Cedex, France; dubova@numer.us.es. doubova@cmapx.polytechnique.fr. This work has been partially supported by D.G.E.S., Spain, Grants PB98–1134.
² 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; axosses@dim.uchile.cl. This work has been partially supported by FONDECYT grants No. 1000955 and 7000955.
³ 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; jppuel@cmapx.polytechnique.fr.

Received: 23 October 2001
Revised: 7 February 2002

Abstract

The results of this paper concern exact controllability to the trajectories for a coupled system of semilinear heat equations. We have transmission conditions on the interface and Dirichlet boundary conditions at the external part of the boundary so that the system can be viewed as a single equation with discontinuous coefficients in the principal part. Exact controllability to the trajectories is proved when we consider distributed controls supported in the part of the domain where the diffusion coefficient is the smaller and if the nonlinear term f(y) grows slower than |y|log^3/2(1+|y|) at infinity. In the proof we use null controllability results for the associate linear system and global Carleman estimates with explicit bounds or combinations of several of these estimates. In order to treat the terms appearing on the interface, we have to construct specific weight functions depending on geometry.