Lemoine, Jerome , MARÍN GAYTE, IRENE, Munch, Arnaud
Si
ESAIM-Control OPtim. Calc. Var.
Article
Científica
1.708
1.015
22/06/2021
000664960200001
The null distributed controllability of the semilinear heat equation partial differential (t)y - Delta y + g(y) = f 1(omega) assuming that g is an element of C-1(DOUBLE-STRUCK CAPITAL R) satisfies the growth condition lim sup(|r|->infinity)g(r)/(|r|ln(3/2)|r|) = 0 has been obtained by Fernandez-Cara and Zuazua (2000). The proof based on a non constructive fixed point theorem makes use of precise estimates of the observability constant for a linearized heat equation. Assuming that g ' is bounded and uniformly Holder continuous on DOUBLE-STRUCK CAPITAL R with exponent p is an element of (0, 1], we design a constructive proof yielding an explicit sequence converging strongly to a controlled solution for the semilinear equation, at least with order 1 + p after a finite number of iterations. The method is based on a least-squares approach and coincides with a globally convergent damped Newton method: it guarantees the convergence whatever be the initial element of the sequence. Numerical experiments in the one dimensional setting illustrate our analysis.
Semilinear heat equation; null controllability; least-squares method