Approximation of exact controls for semi-linear 1D wave equations using a least-squares approach

The exact distributed controllability of the semilinear wave equation $y_{tt}-y_{xx} + g(y)=f \,1_{\omega}$, assuming that $g$ satisfies the growth condition $\vert g(s)\vert /(\vert s\vert \log^{2}(\vert s\vert))\rightarrow 0$ as $\vert s\vert \rightarrow \infty$ and that $g^\prime\in L^\infty_{loc}(\mathbb{R})$ has been obtained by Zuazua in the nineties. The proof based on a Leray-Schauder fixed point argument makes use of precise estimates of the observability constant for a linearized wave equation. It does not provide however an explicit construction of a null control. Assuming that $g^\prime\in L^\infty_{loc}(\mathbb{R})$, that $\sup_{a,b\in \mathbb{R},a\neq b} \vert g^\prime(a)-g^{\prime}(b)\vert/\vert a-b\vert^r<\infty$ for some $r\in (0,1]$ and that $g^\prime$ satisfies the growth condition $\vert g^\prime(s)\vert/\log^{2}(\vert s\vert)\rightarrow 0$ as $\vert s\vert \rightarrow \infty$, we construct an explicit sequence converging strongly to a null control for the solution of the semilinear equation. The method, based on a least-squares approach guarantees the convergence whatever the initial element of the sequence may be. In particular, after a finite number of iterations, the convergence is super linear with rate $1+r$. This general method provides a constructive proof of the exact controllability for the semilinear wave equation.