Approximate controllability of the semilinear reaction-diffusion equation governed by a multiplicative control

In this paper we are concerned with the approximate controllability of a multidimensional semilinear reaction-diffusion equation governed by a multiplicative control, which is locally distributed in the reaction term. For a given initial state we provide sufficient conditions on the desirable state to be approximately reached within an arbitrarily small time interval. Moreover, in the case of a globally supported control, we prove the approximate controllability within any time-interval given in advance which does not depend on the initial and target states. Our approaches are based on linear semigroup theory and some results on uniform approximation with smooth functions.