APPROXIMATE CONTROLLABILITY OF THE SEMILINEAR HEAT-EQUATION

This article is concerned with the study of approximate controllabilit y for the semilinear heat equation in a bounded domain Omega when the control acts on any open and nonempty subset of Omega or on a part of the boundary. In the case of both an internal and a boundary control, the approximate controllability in L(p)(Omega) for 1 less than or equa l to p < + co is proved when the nonlinearity is globally Lipschitz wi th a control in L(infinity). In the case of the interior control, we a lso prove approximate controllability in C-0(Omega). The proof combine s a variational approach to the controllability problem for linear equ ations and a fixed point method. We also prove that the control can be taken to be of ''quasi bang-bang'' form.