An Extension of the Invariance Principle for a Class of Differential Equations with Finite Delay
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Research Article An Extension of the Invariance Principle for a Class of Differential Equations with Finite Delay Marcos Rabelo1 and L. F. C. Alberto2 1 2
Departamento de Matem´atica, Universidade Federal de Pernambuco, UFPE, Recife, PE, Brazil Departamento de Engenharia El´etrica, Universidade de S˜ao Paulo, S˜ao Carlos, SP, Brazil
Correspondence should be addressed to Marcos Rabelo, [email protected] Received 7 October 2010; Accepted 16 December 2010 Academic Editor: Binggen Zhang Copyright q 2010 M. 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. 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 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.
1. Introduction The invariance principle is one of the most important tools to study the asymptotic behavior of differential equations. The first effort to establish invariance principle results for ODEs was likely made by Krasovski˘ı; see 1. Later, other authors have made important contributions to the development of this theory; in particular, the work of LaSalle is of great importance 2, 3. Since then, many versions of the classical invariance principle have been given. For instance, this principle has been successfully extended to differential equations on infinite dimensional spaces, 4–7, including functional differential equations FDEs and, in particular, retarded functional differential equations RFDEs. The great advantage of this principle is the possibility of studying the asymptotic behavior of solutions of differential equations without the explicit knowledge of solutions. For this purpose, the invariance principle supposes the existence of a scalar auxiliary function V satisfying V˙ ≤ 0 and studies the implication of the existence of such function on the ω-limit of solutions.
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Advances in Difference Equations
More recently, the invariance principle was successfully extended to allow the derivative of the scalar function V to be positive in some bounded regions and also to take into account parameter uncertainties. For ordinary differential equations, see 8, 9 and for discrete differential systems, see 10. The main advantage of these extensions is the possibility of applying the invariance theory for a larger class of systems, that is, syst
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