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.

本文讨论了一个热方程的内部零可控性问题，该方程位于带Dirichlet边界条件的有界域上，并受半线性非局部项干扰。我们证明了方程的小局部局部可控性。证明有两个主要论点。首先，我们建立一个\（2 \ times 2 \）反应扩散系统的小时间局部零可控制性，其中第二个方程由抛物线算子\（\ tau \ partial _t-\ sigma \ varDelta \ ），\（\ tau，\ sigma> 0 \）。更准确地说，相对于参数\（（\ tau，\ sigma）\ in（0,1）\ times（1，+ \ infty）\）均匀地获得此可控制性结果。。其次，我们观察到半线性非局部热方程实际上是极限\（（（tau，\ sigma）\ rightarrow（0，+ \ infty）\）中反应扩散系统的渐近推导。最后，我们通过数值模拟说明了这些结果。