The Necessity of the Small-Gain Theorem for Time-Varying and Nonlinear Systems

Jeff S. Shamma*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

32 Scopus citations


It is shown that an appropriately modified version of the small-gain theorem which employs the “asymptotic gain” of a fading-memory system leads to a necessary and sufficient condition for the stable invertibility of certain feedback operators. This modified small-gain theorem is then used to obtain conditions for robust finite-energy input-output stability of 1) nonlinear plants subject to nonlinear time-varying perturbations and 2) linear time-varying plants subject to linear time-varying perturbations. These conditions are shown to be necessary as well as sufficient.

Original languageEnglish (US)
Pages (from-to)1138-1147
Number of pages10
JournalIEEE Transactions on Automatic Control
Issue number10
StatePublished - Oct 1991
Externally publishedYes

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Computer Science Applications
  • Electrical and Electronic Engineering


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