@inbook{a7ab1f6a59ac4ce4b9f8560eed4803eb,
title = "Checkable conditions for contraction after small transients in time and amplitude",
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.",
author = "Michael Margaliot and Tamir Tuller and Sontag, {Eduardo D.}",
note = "Funding Information: We thank Zvi Artstein and George Weiss for helpful comments. EDS{\textquoteright}s work is supported in part by grants NIH 1R01GM100473, AFOSR FA9550-14-1-0060, and ONR N00014-13-1-0074. The research of MM and TT is partly supported by a research grant from the Israeli Ministry of Science, Technology and Space. The research of EDS, MM and TT is also supported by a research grant from the US-Israel Binational Science Foundation. We thank an anonymous reviewer for carefully reading this chapter and providing us several useful comments. Publisher Copyright: {\textcopyright} 2017, Springer International Publishing AG.",
year = "2017",
month = mar,
day = "1",
doi = "10.1007/978-3-319-51298-3_11",
language = "English (US)",
series = "Lecture Notes in Control and Information Sciences",
publisher = "Springer Verlag",
pages = "279--305",
booktitle = "Lecture Notes in Control and Information Sciences",
address = "Germany",
}