Advances in Difference Equations
Volume 2010 (2010), Article ID 496936, 14 pages
doi:10.1155/2010/496936
Research Article

An Extension of the Invariance Principle for a Class of Differential Equations with Finite Delay

Marcos Rabelo1 and L. F. C. Alberto2

1Departamento de Matemática, Universidade Federal de Pernambuco, UFPE, Recife, PE, Brazil
2Departamento de Engenharia Elétrica, Universidade de São Paulo, São Carlos, SP, Brazil

Received 7 October 2010; Accepted 16 December 2010

Academic Editor: Binggen Zhang

Copyright © 2010 Marcos Rabelo and L. F. C. Alberto. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

An extension of the uniform invariance principle for ordinary differential equations with finite delay is developed. The uniform invariance principle allows the derivative of the auxiliary scalar function 𝑉 to be positive in some bounded sets of the state space while the classical invariance principle assumes that 0 x 0 0 3 0 7 𝑉 ≤ 0 . As a consequence, the uniform invariance principle can deal with a larger class of problems. The main difficulty to prove an invariance principle for functional differential equations is the fact that flows are defined on an infinite dimensional space and, in such spaces, bounded solutions may not be precompact. This difficulty is overcome by imposing the vector field taking bounded sets into bounded sets.