On three generalizations of contraction. Sontag, E., Margaliot, M., & Tuller, T. In Proc. IEEE Conf. Decision and Control, Los Angeles, Dec. 2014, pages 1539-1544, 2014. abstract bibtex We introduce three forms of generalized contraction~(GC). Roughly speaking, these are motivated by allowing contraction to take place after small transients in time and/or amplitude. Indeed, contraction is usually used to prove asymptotic properties, like convergence to an attractor or entrainment to a periodic excitation, and allowing initial transients does not affect this asymptotic behavior. We provide sufficient conditions for GC, and demonstrate their usefulness using examples of systems that are not contractive, with respect to any norm, yet are~GC.
@INPROCEEDINGS{14cdc_margaliot_sontag_tuller_cast,
AUTHOR = {E.D. Sontag and M. Margaliot and T. Tuller},
BOOKTITLE = {Proc. IEEE Conf. Decision and Control, Los Angeles, Dec. 2014},
TITLE = {On three generalizations of contraction},
YEAR = {2014},
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PAGES = {1539-1544},
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KEYWORDS = {contractions, contractive systems, stability},
ABSTRACT = {We introduce three forms of generalized
contraction~(GC). Roughly speaking, these are motivated by allowing
contraction to take place after small transients in time and/or
amplitude. Indeed, contraction is usually used to prove asymptotic
properties, like convergence to an attractor or entrainment to a
periodic excitation, and allowing initial transients does not affect
this asymptotic behavior. We provide sufficient conditions for GC,
and demonstrate their usefulness using examples of systems that are
not contractive, with respect to any norm, yet are~GC.}
}
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