On the Liapunov-Krasovskii methodology for the ISS of systems described by coupled delay differential and difference equations

P. Pepe, I. Karafyllis, Z. P. Jiang

Research output: Contribution to journalArticle


The input-to-state stability of time-invariant systems described by coupled differential and difference equations with multiple noncommensurate and distributed time delays is investigated in this paper. Such equations include neutral functional differential equations in Hale's form (which model, for instance, partial element equivalent circuits) and describe lossless propagation phenomena occurring in thermal, hydraulic and electrical engineering. A general methodology for systematically studying the input-to-state stability, by means of Liapunov-Krasovskii functionals, with respect to measurable and locally essentially bounded inputs, is provided. The technical problem concerning the absolute continuity of the functional evaluated at the solution has been studied and solved by introducing the hypothesis that the functional is locally Lipschitz. Computationally checkable LMI conditions are provided for the linear case. It is proved that a linear neutral system in Hale's form with stable difference operator is input-to-state stable if and only if the trivial solution in the unforced case is asymptotically stable. A nonlinear example taken from the literature, concerning an electrical device, is reported, showing the effectiveness of the proposed methodology.

Original languageEnglish (US)
Pages (from-to)2266-2273
Number of pages8
Issue number9
StatePublished - Sep 1 2008



  • Continuous time difference equations
  • Delay differential equations
  • Input-to-state stability
  • Liapunov-Krasovskii functional
  • Neutral systems

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Electrical and Electronic Engineering

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