Multiplicative controllability for semilinear reaction-diffusion equations with finitely many changes of sign

We study the global approximate controllability properties of a one dimensional semilinear reaction-diffusion equation governed via the coefficient of the reaction term. It is assumed that both the initial and target states admit no more than finitely many changes of sign. Our goal is to show that any target state $u^*\in H_0^1 (0,1)$, with as many changes of sign in the same order as the given initial data $u_0\in H^1_0(0,1)$, can be approximately reached in the $L^2 (0,1)$-norm at some time $T>0$. Our method employs shifting the points of sign change by making use of a finite sequence of initial-value pure diffusion problems.