arXiv subjects

Jérôme Lemoine

Publications and source records attributed to Jérôme Lemoine.

3 recordsLinked to original sources

arXiv2023

On the exact boundary controllability of semilinear wave equations

We address the exact boundary controllability of the semilinear wave equation $\partial_{tt}y-Δy + f(y)=0$ posed over a bounded domain $Ω$ of $\mathbb{R}^d$. Assuming that $f$ is continuous and satisfies the condition $\limsup_{\vert r\vert\to \infty} \vert f(r)\vert /(\vert r\vert \ln^p\vert r\vert)\leq β$ for some $β$ small enough and some $p\in [0,3/2)$, we apply the Schauder fixed point theorem to prove the uniform controllability for initial data in $L^2(Ω)\times H^{-1}(Ω)$. Then, assuming that $f$ is in $\mathcal{C}^1(\mathbb{R})$ and satisfies the condition $\limsup_{\vert r\vert\to \infty} \vert f^\prime(r)\vert/\ln^p\vert r\vert\leq β$, we apply the Banach fixed point theorem and exhibit a strongly convergent sequence to a state-control pair for the semilinear equation.

math.AP↗

arXiv2021

Constructive exact control of semilinear 1D heat equations

The exact distributed controllability of the semilinear heat equation $\partial_{t}y-Δy + g(y)=f \,1_ω$ posed over multi-dimensional and bounded domains, assuming that $g\in C^1(\mathbb{R})$ satisfies the growth condition $\limsup_{r\to \infty} g(r)/(\vert r\vert \ln^{3/2}\vert r\vert)=0$ has been obtained by Fernández-Cara and Zuazua in 2000. The proof based on a non constructive fixed point arguments makes use of precise estimates of the observability constant for a linearized heat equation. In the one dimensional setting, assuming that $g^\prime$ does not grow faster than $β\ln^{3/2}\vert r\vert$ at infinity for $β>0$ small enough and that $g^\prime$ is uniformly Hölder continuous on $\mathbb{R}$ with exponent $p\in [0,1]$, we design a constructive proof yielding an explicit sequence converging to a controlled solution for the semilinear equation, at least with order $1+p$ after a finite number of iterations.

math.OC↗

arXiv2021

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

The exact distributed controllability of the semilinear wave equation $\partial_{tt}y-Δy + g(y)=f \,1_ω$ posed over multi-dimensional and bounded domains, assuming that $g\in C^1(\mathbb{R})$ satisfies the growth condition $\limsup_{r\to \infty} g(r)/(\vert r\vert \ln^{1/2}\vert r\vert)=0$ 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 $β\ln^{1/2}\vert r\vert$ at infinity for $β>0$ 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.

math.AP↗