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

2021/01/16 by Lemoine, Jérôme, Münch, Arnaud
#Analysis of PDEs (math.AP) #FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2101.06446

Abstract

The exact distributed controllability of the semilinear wave equation ∂_tty-Δy + g(y)=f 1_ω posed over multi-dimensional and bounded domains, assuming that g∈ C¹(ℝ) satisfies the growth condition \limsup_{r→ ∞} 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^′ does not grow faster than βln^1/2\vert r\vert at infinity for β>0 small enough and that g^′ is uniformly Hölder continuous on ℝ with exponent s∈ (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.

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