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.}
}
Downloads: 0