2016
DOI: 10.4171/jems/605
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Fine scales of decay of operator semigroups

Abstract: Motivated by potential applications to partial differential equations, we develop a theory of fine scales of decay rates for operator semigroups. The theory contains, unifies, and extends several notable results in the literature on decay of operator semigroups and yields a number of new ones. Its core is a new operator-theoretical method of deriving rates of decay combining ingredients from functional calculus, and complex, real and harmonic analysis. It also leads to several results of independent interest.

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Cited by 80 publications
(156 citation statements)
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“…These results are based on the theory of nonuniform stability of semigroups developed in [23,4,6,3]. It should also be noted that the following theorem is not limited to the controller structures used in this paper, but instead it applies to any robust controller for which the closed-loop system satisfies R(iω, A e ) = O(g(|ω|)).…”
Section: If We In Particular Choosementioning
confidence: 99%
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“…These results are based on the theory of nonuniform stability of semigroups developed in [23,4,6,3]. It should also be noted that the following theorem is not limited to the controller structures used in this paper, but instead it applies to any robust controller for which the closed-loop system satisfies R(iω, A e ) = O(g(|ω|)).…”
Section: If We In Particular Choosementioning
confidence: 99%
“…This fundamental characterization of robust controllers was originally presented for finite-dimensional linear systems by Francis and Wonham [15] and Davison [13] in 1970's, and it was later generalized for infinite-dimensional linear systems with finite and infinite-dimensional exosystems in [28,30]. The internal model principle also implies that the robust controllers always tolerate a class of uncertainties and inaccuracies in the parameters G 2 and K and in certain parts of the operator G 1 of the controller (3). This property can be exploited in controller design as it sometimes allows the use of approximations in defining G 1 , G 2 , and K, provided that the internal model property is preserved and the closed-loop system achieves the necessary stability properties.…”
Section: Introductionmentioning
confidence: 99%
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“…The latter are for instance obtained by performing a Fourier transform in the time variable of the damped wave operator ∂ decay, the optimal result was proved by [8] (see also [4] for generalizations) and can be stated as follows (see [1,Proposition 2.4] …”
Section: E(u(t))mentioning
confidence: 99%