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On stability for generalized linear differential equations and applications to impulsive systems

Part of: Stability theory General theory in ordinary differential equations

Published online by Cambridge University Press:  15 February 2023

Claudio A. Gallegos  and
Gonzalo Robledo
Claudio A. Gallegos
Affiliation:
Departamento de Matemáticas, Universidad de Chile, Casilla 653, Santiago, Chile (claudio.gallegos.castro@gmail.com; grobledo@uchile.cl)
Gonzalo Robledo
Affiliation:
Departamento de Matemáticas, Universidad de Chile, Casilla 653, Santiago, Chile (claudio.gallegos.castro@gmail.com; grobledo@uchile.cl)
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Abstract

In this paper, we are interested in investigating notions of stability for generalized linear differential equations (GLDEs). Initially, we propose and revisit several definitions of stability and provide a complete characterization of them in terms of upper bounds and asymptotic behaviour of the transition matrix. In addition, we illustrate our stability results for GLDEs to linear periodic systems and linear impulsive differential equations. Finally, we prove that the well-known definitions of uniform asymptotic stability and variational asymptotic stability are equivalent to the global uniform exponential stability introduced in this article.

Keywords

Generalized ordinary differential equationsKurzweil integralStabilityAsymptotic stabilityImpulsive equations

MSC classification

Primary: 34D20: Stability
Secondary: 34A30: Linear equations and systems, general 34A37: Differential equations with impulses
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 154 , Issue 2 , April 2024 , pp. 381 - 407
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Copyright
Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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