Local null-controllability of a nonlocal semilinear heat equation

TL;DR

This paper proves the small-time local null-controllability of a nonlocal semilinear heat equation on a bounded domain, using reaction-diffusion system controllability and asymptotic analysis, supported by numerical simulations.

Contribution

It introduces a novel approach linking reaction-diffusion system controllability to nonlocal heat equations through asymptotic limits.

Findings

Established controllability of a reaction-diffusion system uniformly over parameters

Derived the nonlocal heat equation as an asymptotic limit of the reaction-diffusion system

Validated theoretical results with numerical simulations

Abstract

This paper deals with the problem of internal null-controllability of a heat equation posed on a bounded domain with Dirichlet boundary conditions and perturbed by a semilinear nonlocal term. We prove the small-time local null-controllability of the equation. The proof relies on two main arguments. First, we establish the small-time local null-controllability of a $2 \times 2$ reaction-diffusion system, where the second equation is governed by the parabolic operator $\tau \partial_t - \sigma \Delta$, $\tau, \sigma > 0$. More precisely, this controllability result is obtained uniformly with respect to the parameters $(\tau, \sigma) \in (0,1) \times (1, + \infty)$. Secondly, we observe that the semilinear nonlocal heat equation is actually the asymptotic derivation of the reaction-diffusion system in the limit $(\tau,\sigma) \rightarrow (0,+\infty)$. Finally, we illustrate these results by numerical simulations.