Stability of Nonlinear Functional Differential Equations by the Contraction Mapping Principle
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Date
2016-08-29
Authors
Ramirez Ibanez, Cesar
Journal Title
Journal ISSN
Volume Title
Publisher
University of Waterloo
Abstract
Fixed point theory has a long history of being used in nonlinear differential
equations, in order to prove existence, uniqueness, or other qualitative properties
of solutions. However, using the contraction mapping principle for stability and
asymptotic stability of solutions is of more recent appearance. Lyapunov functional
methods have dominated the determination of stability for general nonlinear systems
without solving the systems themselves. In particular, as functional differential
equations (FDEs) are more complicated than ODEs, obtaining methods to determine
stability of equations that are difficult to handle takes precedence over analytical
formulas. Applying Lyapunov techniques can be challenging, and the Banach fixed
point method has been shown to yield less restrictive criteria for stability of
delayed FDEs. We will study how to apply the contraction mapping principle to
stability under different conditions to the ones considered by previous authors. We
will first extend a contraction mapping stability result that gives asymptotic
stability of a nonlinear time-delayed scalar FDE which is linearly dominated by the
last state of the system, in order to obtain uniform stability plus asymptotic
stability. We will also generalize to the vector case. Afterwards we do further
extension by considering an impulsively perturbed version of the previous result,
and subsequently we shall use impulses to stabilize an unstable system, under a
contraction method paradigm. At the end we also extend the method to a time
dependent switched system, where difficulties that do not arise in non-switched
systems show up, namely a dwell-time condition, which has already been studied by
previous authors using Lyapunov methods. In this study, we will also deepen
understanding of this method, as well as point out some other difficulties about
using this technique, even for non-switched systems. The purpose is to prompt
further investigations into this method, since sometimes one must consider more than
one aspect other than stability, and having more than one stability criterion might
yield benefits to the modeler.
Description
Keywords
Stability of nonlinear functional differential equations, Stability of nonlinear functional differential equations by a fixed point theorem, stability by the contraction mapping principle, stability of nonlinear systems by Banach contraction theorem