Adilson J. V. Brandao, Enrique Fernandez-Cara,
Paulo M. D. Magalhaes, Marko Antonio Rojas-Medar
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.
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Adilson J. V. Brandão Universidade Federal do ABC - UFABC Santo André, SP, Brazil email: adilson.brandao@ufabc.edu.br |
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| Enrique Fernández-Cara Dpto. E.D.A.N., University of Sevilla Aptdo. 1160, 41080 Sevilla, Spain email: cara@us.es | |
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Paulo M. D. Magalhães DEMAT/ICEB Universidade Federal de Ouro Preto-MG, Brazil email: pmdm@iceb.ufop.br |
| Marko Antonio Rojas-Medar Dpto. Ciencias Básicas, University of Bio-Bio, Campus Fernando May Casilla 447, Chillán, Chile email: marko@ueubiobio.cl |
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