Electron. J. Diff. Eqns., Vol. 2008(2008), No. 164, pp. 1-20.

Theoretical analysis and control results for the FitzHugh-Nagumo equation

Adilson J. V. Brandao, Enrique Fernandez-Cara, Paulo M. D. Magalhaes, Marko Antonio Rojas-Medar

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
  Enrique Fernández-Cara
Dpto. E.D.A.N., University of Sevilla
Aptdo. 1160, 41080 Sevilla, Spain
email: cara@us.es
Paulo M. D. Magalhães
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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