Constructive exact control of semilinear 1D wave equations by a least-squares approach

It has been proved by Zuazua in the nineties that the internally controlled semilinear 1D wave equation $\partial_{tt}y-\partial_{xx}y + g(y)=f 1_{\omega}$,withDirichletboundaryconditions,isexactlycontrollablein$H^1_0(0,1)\cap L^2(0,1)$withcontrols$f\in L^2((0,1)\times(0,T))$,forany$T>0$andanynonemptyopensubset$\omega$of$(0,1)$,assumingthat$g\in $\mathcal{C}$^1(\R)$doesnotgrowfasterthan$\beta\vert x\vert \ln^{2}\vert x\vert$atinfinityforsome$\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 $g^{\mathrm{\prime }}$ does not grow faster than $\beta {\ln }^2\mathrm{\mid }x\mathrm{\mid }$ at infinity for some $\beta >0$ small enough and that $g^{\mathrm{\prime }}$ is uniformly H"older continuous on $\mathbb{R}$ with exponent $s\in [0,1]$, we design a 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.