Asymptotic amplitudes and Cauchy gains: A small-gain principle and an application to inhibitory biological feedback. Sontag, E. Systems Control Lett., 47(2):167–179, 2002.
abstract   bibtex   
The notions of asymptotic amplitude for signals, and Cauchy gain for input/output systems, and an associated small-gain principle, are introduced. These concepts allow the consideration of systems with multiple, and possibly feedback-dependent, steady states. A Lyapunov-like characterization allows the computation of gains for state-space systems, and the formulation of sufficient conditions insuring the lack of oscillations and chaotic behaviors in a wide variety of cascades and feedback loops. An application in biology (MAPK signaling) is worked out in detail.
@ARTICLE{cauchySCL02,
   AUTHOR       = {E.D. Sontag},
   JOURNAL      = {Systems Control Lett.},
   TITLE        = {Asymptotic amplitudes and Cauchy gains: A small-gain 
      principle and an application to inhibitory biological feedback},
   YEAR         = {2002},
   OPTMONTH     = {},
   OPTNOTE      = {},
   NUMBER       = {2},
   PAGES        = {167--179},
   VOLUME       = {47},
   KEYWORDS     = {cyclic feedback systems, small-gain},
   PDF          = {../../FTPDIR/cauchy-gains-scl-as-published.pdf},
   ABSTRACT     = { The notions of asymptotic amplitude for signals, and 
      Cauchy gain for input/output systems, and an associated small-gain 
      principle, are introduced. These concepts allow the consideration of 
      systems with multiple, and possibly feedback-dependent, steady 
      states. A Lyapunov-like characterization allows the computation of 
      gains for state-space systems, and the formulation of sufficient 
      conditions insuring the lack of oscillations and chaotic behaviors in 
      a wide variety of cascades and feedback loops. An application in 
      biology (MAPK signaling) is worked out in detail. }
}
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