Contraction after small transients

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• 作者：Michael Margaliot^a ; ^{michaelm@eng.tau.ac.il" class="auth_mail" title="E-mail the corresponding author}Author Vitae ; Eduardo D. Sontag^b ; ^{eduardo.sontag@rutgers.edu" class="auth_mail" title="E-mail the corresponding author}Author Vitae ; Tamir Tuller^c ; ^{tamirtul@post.tau.ac.il" class="auth_mail" title="E-mail the corresponding author}Author Vitae
• 关键词：Differential analysis ; Contraction ; Stability ; Entrainment ; Phase locking ; Systems biology
• 刊名：Automatica
• 出版年：2016
• 出版时间：May 2016
• 年：2016
• 卷：67
• 期：Complete
• 页码：178-184
• 全文大小：542 K

文摘

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 three 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.