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Revista de Matemática Teoría y Aplicaciones

versão impressa ISSN 1409-2433

Rev. Mat vol.25 no.2 San José Jul./Dez. 2018

http://dx.doi.org/10.15517/rmta.v25i2.33692 

Artículos

Unexistence of limit cycle in an optimal control problem of a population of diabetics

Inexistencia de ciclo límite en un problema de control óptimo de una población con diabetes

Séverine Bernard1 

Ténissia César1 

Silvère P. Nuiro1 

Alain Piétrus1 

1Laboratoire de Mathématiques Informatique et Applications, Université des Antilles, Guadeloupe, France. E-Mail: severine.bernard@univ-antilles.fr, tenissia.cesar@univ-antilles.fr, paul.nuiro@univ-antilles.fr, alain.pietrus@univ-antilles.fr

Abstract

[15]

This paper deals with one of the most important public health problem in the whole world that is diabetes, and more precisely its complications. From a model examining the complications or not of a population of diabetics, we associate a nonlinear optimal control problem. Considering the previous, we prove that the equilibrium state exists and is a saddle point. Moreover, we claim the unexistence of limit cycle in such a population, which is an interesting result concerning this world evil. Then we give some examples for which we characterize the equilibrium state which is not necessarily admissible.

Keywords: two-dimensional optimal control model; limit cycle; equilibrium state; Hopf bifurcation theorem

Resumen

[19]

La diabetes, debido a sus complicaciones, es una de las enfermedades que más problemas plantean en la salud pública actual mundial. En este trabajo se parte de una población de diabéticos con y sin complicaciones y se asocia un problema de control óptimo no lineal que describe la dinámica de la población. Para este modelo se prueba la existencia del estado de equilibrio y que es un punto de ensilladura. Además se obtuvo que no existen ciclos límite, lo que es un resultado importante, dado el problema que se describe. Se presentan ejemplos para los cuales el estado de equilibrio que se caracteriza no es necesariamente admisible.

Palabras clave: modelo de control optimal bi-dimensional; ciclo límite; estado de equilibrio; teorema de bifurcación de Hopf

Mathematics Subject Classification: 49J15, 34H05, 34H20, 90C46, 34C05.

[25]

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Acknowledgments

We would like to thank the anonymous referees for their valuable suggestions and remarks that enabled us to improve the presentation of this manuscript.

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Received: June 21, 2017; Revised: May 07, 2018; Accepted: May 16, 2018

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