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

Abstract

The exact distributed controllability of the semilinear wave equation ${\partial }_{tt}y-\Delta y+g(y)=f{1}_{\omega }$ posed over multi-dimensional and bounded domains, assuming that $g\in C^1(\mathbb{R})$ satisfies the growth condition ${\mathrm{limsup}}_{r\to \infty }g(r)/(|r|{\ln }^{1/2}|r|)=0$ 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 $g^{\prime }$ does not grow faster than $\beta {\ln }^{1/2}|r|$ at infinity for $\beta >0$ small enough and that $g^{\prime }$ is uniformly Hölder 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.

arXiv admin note: substantial text overlap with arXiv:2011.08462, arXiv:2010.14067