The exact distributed controllability of the semilinear heat equation $\partial_{ t}y-\Delta y + g(y)=f \,1_{ \omega}$ posed over multi-dimensional and bounded domains, assuming that $g\in C^1(\mathbb{ R})$ satisfies the growth condition $\limsup_{ r\to \infty} g(r)/(\vert r\vert \ln^{ 3/2}\vert r\vert)=0$ has been obtained by Fern\'andez-Cara and Zuazua in 2000. The proof based on a non constructive fixed point arguments makes use of precise estimates of the observability constant for a linearized heat equation. In the one dimensional setting, assuming that $g^\prime$ does not grow faster than $\beta \ln^{ 3/2}\vert r\vert$ at infinity for $\beta>0$ small enough and that $g^\prime$ is uniformly H\"older continuous on $\mathbb{ R}$ with exponent $p\in [0,1]$, we design a constructive proof yielding an explicit sequence converging to a controlled solution for the semilinear equation, at least with order $1+p$ after a finite number of iterations.
Categories: math.OC