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

Abstract

The exact distributed controllability of the semilinear wave equation ∂tty − ∆y + g(y) = f 1ω posed over multi-dimensional and bounded domains, assuming that g ∈ C 1 (R) satisfies the growth condition lim sup r→∞ 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 ′ does not grow faster than β ln 1/2 |r| 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.