Theoretical Analysis and Control Results for the FitzHugh-Nagumo Equation

In this paper we are concerned with some theoretical questions for the FitzHugh-Nagumo equation. First, we present a simple proof of the existence and uniqueness of strong solution. We also consider an optimal control problem for this system. We prove the existence of optimal state-control pairs and, as an application of the Dubovitski-Milyoutin formalism, we deduce the corresponding optimality system. We also connect the optimal control problem with a controllability question and we construct a sequence of controls that produce solutions that converge strongly to a desired global state. Finally, we present some open questions related to the control of this equation.