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

Abstract

The null distributed controllability of the semilinear heat equation $y_t-\Delta y+g(y)=f{1}_{\omega }$, assuming that $g$ satisfies the growth condition $g(s)/(|s|{\log }^{3/2}(1+|s|))\to 0$ as $|s|\to \infty$ and that $g^{\prime }\in L_{loc}^{\infty }(\mathbb{R})$ has been obtained by Fernández-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 $g^{\prime }\in W^{s,\infty }(\mathbb{R})$ for one $s\in (0,1]$, we construct an explicit sequence converging strongly to a null control for the solution of the semilinear 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.