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Strong input-to-state stability for infinite-dimensional linear systems

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Abstract

This paper deals with strong versions of input-to-state stability for infinite-dimensional linear systems with an unbounded control operator. We show that strong input-to-state stability with respect to inputs in an Orlicz space is a sufficient condition for a system to be strongly integral input-to-state stable with respect to bounded inputs. In contrast to the special case of systems with exponentially stable semigroup, the converse fails in general.

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Notes

  1. We emphasize that a finite-dimensional input space does not imply that B is a bounded operator since in our setting the range of B may exceed the state space; see Sect. 2.

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Acknowledgements

The authors would like to thank Birgit Jacob for valuable discussions and helpful comments on the manuscript. They are also very grateful to the anonymous referees for their careful reading of the manuscript and their suggestions for improvements.

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Authors and Affiliations

  1. School of Mathematics and Natural Sciences, University of Wuppertal, Gaußstraße 20, 42119, Wuppertal, Germany

    Robert Nabiullin

  2. Department of Mathematics, Center for Optimization and Approximation, University of Hamburg, Bundesstraße 55, 20146, Hamburg, Germany

    Felix L. Schwenninger

Authors
  1. Robert Nabiullin
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  2. Felix L. Schwenninger
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Corresponding author

Correspondence to Robert Nabiullin.

Additional information

RN is supported by Deutsche Forschungsgemeinschaft (Grant JA 735/12-1).

FLS is supported by Deutsche Forschungsgemeinschaft (Grant RE 2917/4-1).

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Nabiullin, R., Schwenninger, F.L. Strong input-to-state stability for infinite-dimensional linear systems. Math. Control Signals Syst. 30, 4 (2018). https://doi.org/10.1007/s00498-018-0210-8

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  • DOI: https://doi.org/10.1007/s00498-018-0210-8

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