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 2×2 reaction-diffusion system, where the second equation is governed by the parabolic operator τ ∂ t − σ∆, τ, σ > 0. More precisely, this controllability result is obtained uniformly with respect to the parameters (τ, σ) ∈ (0, 1) × (1, +∞). Secondly, we observe that the semilinear nonlocal heat equation is actually the asymptotic derivation of the reaction-diffusion system in the limit (τ, σ) → (0, +∞). Finally, we illustrate these results by numerical simulations.