Contraction after small transients. (May 2016)

• Record Type: Journal Article
• Authors: Margaliot, Michael; Sontag, Eduardo D.; Tuller, Tamir
• Is Part Of: Automatica. Volume 67(2016)
• Page Start: 178
• Page End: 184
• Publication Date: 2016-05
• Subjects: Differential analysis -- Contraction -- Stability -- Entrainment -- Phase locking -- Systems biology; Automatic control -- Periodicals; Automation -- Periodicals; 629.805
• DOI: 10.1016/j.automatica.2016.01.018
• Languages: English
• ISSNs: 0005-1098

Abstract

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 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.