Constructive exact control of semilinear 1D heat equations

The exact distributed controllability of the semilinear heat equation ${\mathrm{\partial }}_{t}y-\mathrm{\Delta }y+g(y)=f{1}_{\omega }$ posed over multi-dimensional and bounded domains, assuming that $g\in C^1(\mathbb{R})$ satisfies the growth condition ${\mathrm{lim\hspace{0.17em}sup}}_{r\to \mathrm{\infty }}g(r)\mathrm{/}(\mathrm{\mid }r\mathrm{\mid }{\ln }^{3\mathrm{/}2}\mathrm{\mid }r\mathrm{\mid })=0$ 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 $g^{\mathrm{\prime }}$ does not grow faster than $\beta \ln^{3/2}\vert r\vert$atinfinityfor$\beta>0$smallenoughandthat$g^\prime$isuniformlyH\ddot{o}ldercontinuouson$$\mathbb{R}$$withexponent$p\in [0,1]$, 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.