Checkable conditions for contraction after small transients in time and amplitude

Michael Margaliot*, Tamir Tuller, Eduardo D. Sontag

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

11 Scopus citations

Abstract

Contraction theory is a powerful tool for proving asymptotic properties of nonlinear dynamical systems including convergence to an attractor and entrainment to a periodic excitation. We consider generalizations of contraction with respect to a norm that allow contraction to take place after small transients in time and/or amplitude. These generalized contractive systems (GCSs) are useful for several reasons. First, we show that there exist simple and checkable conditions guaranteeing that a system is a GCS, and demonstrate their usefulness using several models from systems biology. Second, allowing small transients does not destroy the important asymptotic properties of contractive systems like convergence to a unique equilibrium point, if it exists, and entrainment to a periodic excitation. Third, in some cases as we change the parameters in a contractive system it becomes a GCS just before it looses contractivity with respect to a norm. In this respect, generalized contractivity is the analogue of marginal stability in Lyapunov stability theory.

Original languageEnglish
Title of host publicationLecture Notes in Control and Information Sciences
PublisherSpringer Verlag
Pages279-305
Number of pages27
DOIs
StatePublished - 1 Mar 2017

Publication series

NameLecture Notes in Control and Information Sciences
Volume473
ISSN (Print)0170-8643

Funding

FundersFunder number
National Institutes of Health1R01GM100473
Office of Naval ResearchN00014-13-1-0074
Air Force Office of Scientific ResearchFA9550-14-1-0060
Ministry of Science, Technology and Space
United States-Israel Binational Science Foundation

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