Local Null-Controllability of a Nonlocal Semilinear Heat Equation

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 $$\tau \partial _t - \sigma \varDelta $$, $$\tau , \sigma > 0$$. 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.