Contraction after small transients. Margaliot, M., Sontag, E., & Tuller, T. Automatica, 67:178-184, 2016. abstract bibtex 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 introduce three new forms of generalized contraction (GC) that are motivated by allowing contraction to take place after small transients in time and/or amplitude. These forms of GC are useful for several reasons. First, allowing small transients does not destroy the asymptotic properties provided by standard contraction. Second, in some cases as we change the parameters in a contractive system it becomes a GC just before it looses contractivity. In this respect, GC is the analogue of marginal stability in Lyapunov stability theory. We provide checkable sufficient conditions for GC, and demonstrate their usefulness using several models from systems biology that are not contractive, with respect to any norm, yet are GC.
@ARTICLE{margaliot_sontag_tuller_cast_2015,
AUTHOR = {M. Margaliot and E.D. Sontag and T. Tuller},
JOURNAL = {Automatica},
TITLE = {Contraction after small transients},
YEAR = {2016},
OPTMONTH = {},
OPTNOTE = {},
OPTNUMBER = {},
PAGES = {178-184},
VOLUME = {67},
KEYWORDS = {entrainment, nonlinear systems, stability, contractions,
contractive systems},
PDF = {../../FTPDIR/margaliot_sontag_tuller_automatica2016.pdf},
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 introduce three new forms of generalized contraction (GC) that are
motivated by allowing contraction to take place after small
transients in time and/or amplitude. These forms of GC are useful for
several reasons. First, allowing small transients does not destroy
the asymptotic properties provided by standard contraction. Second,
in some cases as we change the parameters in a contractive system it
becomes a GC just before it looses contractivity. In this respect, GC
is the analogue of marginal stability in Lyapunov stability theory.
We provide checkable sufficient conditions for GC, and demonstrate
their usefulness using several models from systems biology that are
not contractive, with respect to any norm, yet are GC.}
}
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