Constructive exact control of semilinear 1D heat equations

Abstract

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 ${\mathrm{limsup}}_{r\to \infty }g(r)/(|r|{\ln }^{3/2}|r|)=0$ 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. In the one dimensional setting, assuming that $g^{\prime }$ does not grow faster than $\beta {\ln }^{3/2}|r|$ at infinity for $\beta >0$ small enough and that $g^{\prime }$ is uniformly Hölder 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.

arXiv admin note: text overlap with arXiv:2008.12656, arXiv:2101.06446