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 $\tau {\partial }_t-\sigma \Delta$, $\tau ,\sigma >0$. 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 $(\tau ,\sigma )\to (0,+\infty )$. Finally, we illustrate these results by numerical simulations.