Exact controllability of a semilinear reaction-diffusion equation governed by a bilinear control

This paper investigates the exact controllability of a semilinear reaction-diffusion equation governed by a multiplicative control. Here, the reaction term leads to an unbounded control operator. We will first study the well-posedness and then proceed to the question of exact steering for the system at hand. Our approaches are based on a fixed point argument and the linearization method. The obtained results are further analysed through a numerical example.