Electronic Journal of Differential Equations,
Vol. 2008(2008), No. 164, pp. 1-20.
Title: Theoretical analysis and control results for the
FitzHugh-Nagumo equation
Authors: Adilson J. V. Brandao (Univ. Federal do ABC, Brazil)
Enrique Fernandez-Cara (Univ. of Sevilla, Spain)
Paulo M. D. Magalhaes (Univ. Federal de Ouro Preto, Brazil)
Marko Antonio Rojas-Medar (Univ. of Bio-Bio, Chile)
Abstract:
In this paper we are concerned with some theoretical questions for the
FitzHugh-Nagumo equation.
First, we recall the system, we briefly explain the meaning of the
variables and we present a simple proof of the existence and
uniqueness of strong solution.
We also consider an optimal control problem for this system.
In this context, the goal is to determine how can we act on the
system in order to get good properties.
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 desired states.
This provides a strategy to make the system behave as desired.
Finally, we present some open questions related to the control of this
equation.
Submitted November 13, 2007. Published December 23, 2008.
Math Subject Classifications: 35B37, 49J20, 93B05.
Key Words: Optimal control; controllability; FitzHugh-Nagumo equation;
Dubovitski-Milyoutin.
An addendum as attached on July 8, 2009. It clarifies a controllability result. See last page of this article.