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We study the global approximate controllability properties of a onedimensional semilinear reaction-diffusion equation governed via the coefficientof the reaction term. It is assumed that both the initial and target statesadmit no more than finitely many changes of sign. Our goal is to show that anytarget state $ u^*\in H_0^1 (0,1)$, with as many changes of sign in the sameorder as the given initial data $ u_0\in H^1_0(0,1)$, can be approximatelyreached in the $ L^2 (0,1)$-norm at some time $T>0$. Our method employsshifting the points of sign change by making use of a finite sequence ofinitial-value pure diffusion problems.

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