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The ISS framework for time-delay systems: a survey

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Abstract

At the occasion of Eduardo D. Sontag’s 70th birthday, we provide here an overview of the tools available to study input-to-state stability (ISS) and related notions for time-delay systems. After a hopefully pedagogical presentation of the main differences with respect to the finite-dimensional theory, we review basic stability concepts for input-free time-delay systems, as well as instruments to guarantee them in practice, including the Lyapunov–Krasovskii, Lyapunov–Razumikhin, and Halanay approaches. We then consider the influence of inputs through the notions of ISS, integral ISS, and input-to-output stability and provide both Lyapunov-like and solutions-based characterizations of these properties. We also show how these notions can be helpful for the stability analysis of interconnected systems, whether in cascade or in feedback form. We finally provide a list of questions which remain open until now.

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Notes

  1. Which coincides with the derivative of \(t\mapsto x(t)\) at each time t where x is differentiable.

  2. This implication implicitly imposes that the system is forward complete

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Acknowledgements

The authors would like to warmly thank Gökhan Göksu and Epiphane Loko for their careful reading of this survey.

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Authors and Affiliations

  1. Universitè Paris-Saclay, CNRS, CentraleSupèlec, Laboratoire des signaux et systèmes, 91190, Gif-sur-Yvette, France

    Antoine Chaillet

  2. Mathematics Department, National Technical University of Athens, 15780, Athens, Greece

    Iasson Karafyllis

  3. Information Engineering, Computer Science, and Mathematics Department, University of L’Aquila, 67100, L’Aquila, Italy

    Pierdomenico Pepe

  4. Mathematical Sciences Department, Florida Atlantic University, 33431, Boca Raton, FL, Italy

    Yuan Wang

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  1. Antoine Chaillet
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This paper is dedicated to Eduardo Sontag on the occasion of his 70th birthday.

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This survey paper is dedicated to E.D. Sontag, at the occasion of his 70th birthday, as a humble testimony of his fundamental and breakthrough contributions to the field of stability analysis of nonlinear systems and as a token of his inspiring work

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Chaillet, A., Karafyllis, I., Pepe, P. et al. The ISS framework for time-delay systems: a survey. Math. Control Signals Syst. 35, 237–306 (2023). https://doi.org/10.1007/s00498-023-00341-w

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