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Stability of infinite-dimensional sampled-data systems


Authors: Hartmut Logemann, Richard Rebarber and Stuart Townley
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
Published electronically: April 25, 2003
MathSciNet review: 1974689
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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

DOI: https://doi.org/10.1090/S0002-9947-03-03142-8
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).
Article copyright: © Copyright 2003 American Mathematical Society

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