Approximate controllability for semilinear heat equations with globally Lipschitz nonlinearities

We consider the semilinear heat equation involving gradient terms in a boun ded domain of R-n. It is assumed that the non-linearity is globally Lipschi tz. We prove that the system is approximately controllable when the control acts on abounded subset of the domain. The proof uses a variant of a class ical fixed point method and is a simpler alternative to the earlier proof e xisting in the literature by means of the penalization of an optimal contro l problem. We also prove that the control may be built; so that, in additio n to the approximate controllability requirement, it ensures that the state reaches exactly a finite number of constraints.