Approximate and exact controllability of a reaction-diffusion equation governed by bilinear control

Our goal in this paper is to investigate the global controllability for a reaction-diffusion equation governed in a bounded domain Ω⊂Rn by a multiplicative control in the reaction term. We will show that, under some sufficient conditions on a given couple of initial and desirable states, the considered system can be steered in L2(Ω) from its initial state into any neighborhood of the given desirable target state by means of a static control. This result is further applied to discuss the exact controllability properties for the one dimensional reaction-diffusion equation. Our approaches are based on linear semigroup theory and the null controllability of linear systems with small initial states.