Exact controllability of semilinear heat equations through a constructive approach

The exact distributed controllability of the semilinear heat equation \begin{document}$ \partial_{t}y-\Delta y + f(y) = v \, 1_{\omega} $\end{document} posed over multi-dimensional and bounded domains, assuming that \begin{document}$ f $\end{document} is locally Lipschitz continuous and satisfies the growth condition \begin{document}$ \limsup_{| r|\to \infty} | f(r)| /(| r| \ln^{3/2}| r|)\leq \beta $\end{document} for some \begin{document}$ \beta $\end{document} small enough has been obtained by Fernández-Cara and Zuazua in 2000. The proof based on a non constructive fixed point arguments makes use of precise estimates of the observability constant for a linearized heat equation. Under the same assumption, by introducing a different fixed point application, we present a different and somewhat simpler proof of the exact controllability, which is not based on the cost of observability of the heat equation with respect to potentials. Then, assuming that \begin{document}$ f $\end{document} is locally Lipschitz continuous and satisfies the growth condition \begin{document}$ \limsup_{| r|\to \infty} | f^\prime(r)|/\ln^{3/2}| r|\leq \beta $\end{document} for some \begin{document}$ \beta $\end{document} small enough, we show that the above fixed point application is contracting yielding a constructive method to compute the controls for the semilinear equation. Numerical experiments illustrate the results.