Approximation of null controls for semilinear heat equations using a least-squares approach

The null distributed controllability of the semilinear heat equation y t − ∆y + g(y) = f 1 ω , assuming that g satisfies the growth condition g(s)/(|s| log 3/2 (1 + |s|)) → 0 as |s| → ∞ and that g ∈ L ∞ loc (R) has been obtained by Fernandez-Cara and Zuazua in 2000. The proof based on a fixed point argument makes use of precise estimates of the observability constant for a linearized heat equation. It does not provide however an explicit construction of a null control. Assuming that g ∈ W s,∞ (R) for one s ∈ (0, 1], we construct an explicit sequence converging strongly to a null control for the solution of the semi-linear equation. The method, based on a least-squares approach, generalizes Newton type methods and guarantees the convergence whatever be the initial element of the sequence. In particular, after a finite number of iterations, the convergence is super linear with a rate equal to 1 + s. Numerical experiments in the one dimensional setting support our analysis.