Constructive exact control of semilinear 1D heat equations

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