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Input-to-state stability of infinite-dimensional control systems

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Abstract

We develop tools for investigation of input-to-state stability (ISS) of infinite-dimensional control systems. We show that for certain classes of admissible inputs, the existence of an ISS-Lyapunov function implies the ISS of a system. Then for the case of systems described by abstract equations in Banach spaces, we develop two methods of construction of local and global ISS-Lyapunov functions. We prove a linearization principle that allows a construction of a local ISS-Lyapunov function for a system, the linear approximation of which is ISS. In order to study the interconnections of nonlinear infinite-dimensional systems, we generalize the small-gain theorem to the case of infinite-dimensional systems and provide a way to construct an ISS-Lyapunov function for an entire interconnection, if ISS-Lyapunov functions for subsystems are known and the small-gain condition is satisfied. We illustrate the theory on examples of linear and semilinear reaction-diffusion equations.

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Acknowledgments

This research was funded by the German Research Foundation (DFG) as a part of Collaborative Research Centre 637 “Autonomous Cooperating Logistic Processes—A Paradigm Shift and its Limitations”. We thank the anonymous reviewers for their comments and suggestions, which led to improvements in the presentation of the results.

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Authors and Affiliations

  1. Department of Civil Engineering, University of Applied Sciences Erfurt, Erfurt, Germany

    Sergey Dashkovskiy

  2. Faculty of Mathematics and Computer Science, University of Bremen, Bibliothekstraße 1, 28334 , Bremen, Germany

    Andrii Mironchenko

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  1. Sergey Dashkovskiy
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  2. Andrii Mironchenko
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Correspondence to Andrii Mironchenko.

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Dashkovskiy, S., Mironchenko, A. Input-to-state stability of infinite-dimensional control systems. Math. Control Signals Syst. 25, 1–35 (2013). https://doi.org/10.1007/s00498-012-0090-2

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  • DOI: https://doi.org/10.1007/s00498-012-0090-2

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