Constructive exact controls for semi-linear wave equations

The exact distributed controllability of the semi-linear wave equation $\partial_{tt} y - \Delta y + g(y) = f1_\omega$ posed over multi-dimensional and bounded domains, assuming that $g \in \mathcal{C}^1 (\mathbb{R})$ satisfies the growth condition $\lim \sup_{{\lvert r \rvert} \to \infty} g(r) / ({\lvert r \rvert} \ln^{1/2} {\lvert r \rvert}) = 0$ 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 the derivative of $g$ does not grow faster than $\beta \ln^{1/2} {\lvert r \rvert}$ at infinity for $\beta \gt 0$ small enough and 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 semi-linear equation, at least with order $1+s$ after a finite number of iterations. Numerical experiments in the two-dimensional case illustrate the results. This work extends to a multi-dimensional case, enriches with additional results and completes with some numerical experiments the study in 2021 by Münch and Trélat devoted to the one-dimensional situation.

Keywords

semilinear wave equation, exact controllability, least-squares approach

2010 Mathematics Subject Classification

Primary 35L71, 49M15. Secondary 93E24.

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Dedicated to the memory of Professor Roland Glowinski

Received 24 November 2022

Published 14 November 2023