TY - CHAP

T1 - Checkable conditions for contraction after small transients in time and amplitude

AU - Margaliot, Michael

AU - Tuller, Tamir

AU - Sontag, Eduardo D.

N1 - Publisher Copyright:
© 2017, Springer International Publishing AG.

PY - 2017/3/1

Y1 - 2017/3/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85017002820&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-51298-3_11

DO - 10.1007/978-3-319-51298-3_11

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AN - SCOPUS:85017002820

T3 - Lecture Notes in Control and Information Sciences

SP - 279

EP - 305

BT - Lecture Notes in Control and Information Sciences

PB - Springer Verlag

ER -