Constructive proof of the exact controllability for semi-linear wave equations

The exact distributed controllability of the semilinear wave equation ${\mathrm{\partial }}_{tt}y-\mathrm{\Delta }y+g(y)=f{1}_{\omega }$ posed over multi-dimensional and bounded domains, assuming that $g\in C^1(\mathbb{R})$ satisfies the growth condition ${\mathrm{lim\hspace{0.17em}sup}}_{r\to \mathrm{\infty }}g(r)\mathrm{/}(\mathrm{\mid }r\mathrm{\mid }{\ln }^{1\mathrm{/}2}\mathrm{\mid }r\mathrm{\mid })=0$ has been obtained by Fu, Yong and Zhang in 2007. The proof based on a non constructive Leray-Schauder fixed point theorem makes use of precise estimates of the observability constant for a linearized wave equation. Assuming that $g^{\mathrm{\prime }}$ does not grow faster than $\beta {\ln }^{1\mathrm{/}2}\mathrm{\mid }r\mathrm{\mid }$ at infinity for $\beta >0$ small enough and that $g^{\mathrm{\prime }}$ is uniformly H"older continuous on $\mathbb{R}$ with exponent $s\in (0,1]$, we design a constructive proof yielding an explicit sequence converging to a controlled solution for the semilinear equation, at least with order $1+s$ after a finite number of iterations.