Constructive exact controls for semi-linear wave equations

The exact distributed controllability of the semi-linear wave equation partial derivative(tt)y -Delta y + g(y) = f1(omega) posed over multi-dimensional and bounded domains, assuming that g is an element of C-1(R) satisfies the growth condition lim sup(| r|->infinity) 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 the derivative of g does not grow faster than beta ln(1/2) |r| at infinity for beta > 0 small enough and is uniformly Holder continuous on R with exponent s is an element of (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 M<spacing diaeresis>unch and Tr ' elat devoted to the one-dimensional situation.