LAUSR

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.