On delay-derivative-dependent stability of systems with fast-varying delays

Eugenii Shustin, Emilia Fridman*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


Stability of linear systems with uncertain bounded time-varying delays is studied under the assumption that the nominal delay values are not equal to zero. An input-output approach to stability of such systems is known to be based on the bound of the L2-norm of a certain integral operator. There exists a bound on this operator norm in two cases: in the case where the delay derivative is not greater than 1 and in the case without any constraints on the delay derivative. In the present note we fill the gap between the two cases by deriving a tight operator bound which is an increasing and continuous function of the delay derivative upper bound d ≥ 1. For d → ∞ the new bound corresponds to the second case and improves the existing bound. As a result, for the first time, delay-derivative-dependent frequency domain and time domain stability criteria are derived for systems with the delay derivative greater than 1.

Original languageEnglish
Pages (from-to)1649-1655
Number of pages7
Issue number9
StatePublished - Sep 2007


  • Input-output approach
  • L-norm
  • Stability
  • Time-varying delay


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