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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Stability of infinite-dimensional sampled-data systems
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by Hartmut Logemann, Richard Rebarber and Stuart Townley PDF
Trans. Amer. Math. Soc. 355 (2003), 3301-3328 Request permission

Abstract:

Suppose that a static-state feedback stabilizes a continuous-time linear infinite-dimensional control system. We consider the following question: if we construct a sampled-data controller by applying an idealized sample-and-hold process to a continuous-time stabilizing feedback, will this sampled-data controller stabilize the system for all sufficiently small sampling times? Here the state space $X$ and the control space $U$ are Hilbert spaces, the system is of the form $\dot x(t) = Ax(t) + Bu(t)$, where $A$ is the generator of a strongly continuous semigroup on $X$, and the continuous time feedback is $u(t) = Fx(t)$. The answer to the above question is known to be “yes” if $X$ and $U$ are finite-dimensional spaces. In the infinite-dimensional case, if $F$ is not compact, then it is easy to find counterexamples. Therefore, we restrict attention to compact feedback. We show that the answer to the above question is “yes”, if $B$ is a bounded operator from $U$ into $X$. Moreover, if $B$ is unbounded, we show that the answer “yes” remains correct, provided that the semigroup generated by $A$ is analytic. We use the theory developed for static-state feedback to obtain analogous results for dynamic-output feedback control.
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Additional Information
  • Hartmut Logemann
  • Affiliation: Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom
  • Email: hl@maths.bath.ac.uk
  • Richard Rebarber
  • Affiliation: Department of Mathematics and Statistics, University of Nebraska-Lincoln, Lincoln, Nebraska 68588-0323
  • Email: rrebarbe@math.unl.edu
  • Stuart Townley
  • Affiliation: School of Mathematical Sciences, University of Exeter, Exeter, EX4 4QE, United Kingdom
  • Email: townley@maths.ex.ac.uk
  • Received by editor(s): December 21, 2000
  • Received by editor(s) in revised form: February 21, 2002
  • Published electronically: April 25, 2003
  • Additional Notes: This work was supported by NATO (Grant CRG 950179) and by the National Science Foundation (Grant DMS-9623392).
  • © Copyright 2003 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 355 (2003), 3301-3328
  • MSC (2000): Primary 34G10, 47A55, 47D06, 93C25, 93C57, 93D15
  • DOI: https://doi.org/10.1090/S0002-9947-03-03142-8
  • MathSciNet review: 1974689