  The exact distributed controllability of the semilinear heat equation
$\partial_{t}y-\Delta y + g(y)=f \,1_{\omega}$ posed over multi-dimensional and
bounded domains, assuming that $g\in C^1(\mathbb{R})$ satisfies the growth
condition $\limsup_{r\to \infty} g(r)/(\vert r\vert \ln^{3/2}\vert r\vert)=0$
has been obtained by Fern\'andez-Cara and Zuazua in 2000. The proof based on a
non constructive fixed point arguments makes use of precise estimates of the
observability constant for a linearized heat equation. In the one dimensional
setting, assuming that $g^\prime$ does not grow faster than $\beta
\ln^{3/2}\vert r\vert$ at infinity for $\beta>0$ small enough and that
$g^\prime$ is uniformly H\"older continuous on $\mathbb{R}$ with exponent $p\in
[0,1]$, we design a constructive proof yielding an explicit sequence converging
to a controlled solution for the semilinear equation, at least with order $1+p$
after a finite number of iterations.
