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Nonautonomous Linear Hamiltonian Systems Read chapter Read first chapter

2016 | OriginalPaper | Chapter

5. Weak Disconjugacy for Linear Hamiltonian Systems

Authors : Russell Johnson, Rafael Obaya, Sylvia Novo, Carmen Núñez, Roberta Fabbri

Published in: Nonautonomous Linear Hamiltonian Systems: Oscillation, Spectral Theory and Control

Publisher: Springer International Publishing

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Abstract

The analysis of nonautonomous linear Hamiltonian systems with the disconjugacy property is a classical branch of the theory of linear ODEs. One of the most interesting consequences of this property is the existence of principal functions, which constitute an extension of the concept of Weyl functions to many situations where exponential dichotomy is lacking. In this chapter, the disconjugacy property is relaxed to the so-called weak disconjugacy, which can be characterized in terms of the property of nonoscillation of the system. In what follows, a family of systems is considered. It is proved that the so-called uniform weak disconjugacy of all the systems suffices to ensure the existence of the principal functions, whose main properties are then described. And it is shown that this setting is less restrictive than that of disconjugacy. In the rest of the chapter, relations are established between the characteristics of the principal functions for a given family and: (1) the properties of the Lyapunov exponents, (2) the properties of the rotation number, and (3) the existence of exponential dichotomy. The consequences of the analysis presented here will be fundamental in the developments of the remaining chapters of the book.

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previous chapter The Weyl Functions
next chapter Nonautonomous Control Theory: Linear Regulator Problem and the Kalman–Bucy Filter
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Metadata
Title
Weak Disconjugacy for Linear Hamiltonian Systems
Authors
Russell Johnson
Rafael Obaya
Sylvia Novo
Carmen Núñez
Roberta Fabbri
Copyright Year
2016
Book
Nonautonomous Linear Hamiltonian Systems: Oscillation, Spectral Theory and Control

Print ISBN: 978-3-319-29023-2

Electronic ISBN: 978-3-319-29025-6

Copyright Year: 2016

DOI
https://doi.org/10.1007/978-3-319-29025-6_5