Abstract The FitzHugh-Nagumo (FHN) equation is a simplified model of a nerve axon. We explore the controllability of this model using a localized interior control only for the first equation. The Linearized system is not null controllable using a localized interior control since the spectrum of the linearized system has an accumulation point though it is approximate controllable. We show that the solution of the FHN equation fails to be globally approximate controllable in a given time. But it is possible to move from any steady state to any other steady state arbitrarily close after some appropriate time by a localized interior control, provided that both steady states are in the same connected component of the set of steady states. Finally we make some additional remarks and comments and we mention some open questions for our system. For the sake of completeness, we give the details of the existence, uniqueness and uniform bound of the solution in Appendix.

中文翻译：

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一维 FitzHugh-Nagumo 方程的近似可控性

摘要 FitzHugh-Nagumo (FHN) 方程是神经轴突的简化模型。我们仅对第一个方程使用局部内部控制来探索该模型的可控性。使用局部内部控制的线性化系统不是零可控的，因为线性化系统的频谱虽然是近似可控的，但它有一个累积点。我们表明 FHN 方程的解不能在给定时间内全局近似可控。但是有可能通过局部内部控制在适当的时间后从任何稳态移动到任意关闭的任何其他稳态，前提是两个稳态都在稳态集的同一个连通分量中。最后，我们提出了一些额外的评论和评论，并为我们的系统提出了一些未解决的问题。