Necessary and sufficient small gain conditions for integral input-to-state stable systems: A Lyapunov perspective

Hiroshi Ito, Zhong Ping Jiang

Research output: Contribution to journalArticlepeer-review


This paper is concerned with conditions for the stability of interconnected nonlinear systems consisting of integral input-to-state stable(iISS) systems with external inputs. The treatment of iISS and input-to-state stable (ISS) systems is unified. Both necessary conditions and sufficient conditions are investigated using a Lyapunov formulation. In the presence of model uncertainty, this paper proves that, for the stability of the interconnected system, at least one subsystem is necessarily ISS which is a stronger stability property in the set of iISS. The necessity of a small-gain-type property is also demonstrated. This paper proposes a common form of smooth Lyapunov functions which can establish the iISS and the ISS of the interconnection comprising iISS and ISS subsystems whenever the small-gain-type condition is satisfied. The result covers situations more general than the earlier study and removes technical conditions assumed in the previous literature. Global asymptotic stability is discussed as a special case.

Original languageEnglish (US)
Pages (from-to)2389-2404
Number of pages16
JournalIEEE Transactions on Automatic Control
Issue number10
StatePublished - 2009


  • Integral input-to-state stability (iISS)
  • Lyapunov function
  • Necessary condition
  • Nonlinear interconnected systems
  • Small gain condition

ASJC Scopus subject areas

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


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