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

TL;DR

This paper introduces a least-squares iterative method to explicitly approximate null controls for semilinear heat equations, extending previous theoretical results and providing practical convergence guarantees with numerical validation.

Contribution

It develops a novel least-squares based iterative approach to explicitly construct null controls for semilinear heat equations, with proven convergence properties.

Findings

Method guarantees convergence regardless of initial guess

Achieves super linear convergence rate of 1+s

Numerical experiments confirm theoretical results

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 $g(s)/(\vert s\vert \log^{3/2}(1+\vert s\vert))\rightarrow 0$ as $\vert s\vert \rightarrow \infty$ and that $g^\prime\in L^\infty_{loc}(\mathbb{R})$ has been obtained by Fern\'andez-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.