  It has been proved by Zuazua in the nineties that the internally controlled
semilinear 1D wave equation $\partial_{tt}y-\partial_{xx}y + g(y)=f
1_{\omega}$, with Dirichlet boundary conditions, is exactly controllable in
$H^1_0(0,1)\cap L^2(0,1)$ with controls $f\in L^2((0,1)\times(0,T))$, for any
$T>0$ and any nonempty open subset $\omega$ of $(0,1)$, assuming that $g\in
\mathcal{C}^1(\R)$ does not grow faster than $\beta\vert x\vert \ln^{2}\vert
x\vert$ at infinity for some $\beta>0$ small enough. The proof, based on the
Leray-Schauder fixed point theorem, is however not constructive. In this
article, we design a constructive proof and algorithm for the exact
controllability of semilinear 1D wave equations. Assuming that $g^\prime$ does
not grow faster than $\beta \ln^{2}\vert x\vert$ at infinity for some $\beta>0$
small enough and that $g^\prime$ is uniformly H\"older continuous on $\R$ with
exponent $s\in[0,1]$, we design a least-squares algorithm 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.
