Asymptotic Behaviour of Coupled Systems in Discrete and Continuous Time

Paunonen, Lassi; Seifert, David

doi:10.1007/s10884-016-9547-1

Cited by 5 publications

(3 citation statements)

References 15 publications

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“…When (4.3) holds we call φ the characteristic function of our system. The class of coupled systems possessing a characteristic function was first studied in [68], with further developments in [69,71]. We let 1 ≤ p ≤ ∞ and rewrite the system (4.4) in the form of an abstract Cauchy problem,…”

Section: Infinite Systems or Coupled Ordinary Differential Equationsmentioning

confidence: 99%

Some developments around the Katznelson–Tzafriri theorem

Batty

Seifert

2022

Acta Sci. Math. (Szeged)

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This paper is a survey article on developments arising from a theorem proved by Katznelson and Tzafriri in 1986 showing that $${\rm lim}_{n \rightarrow \infty} ||T^{n}(I-T)|| = 0$$ lim n → ∞ | | T n ( I - T ) | | = 0 if T is a power-bounded operator on a Banach space and $$\sigma (T)\cap \mathbb{T} \subseteq\{1\}$$ σ ( T ) ∩ T ⊆ { 1 } . Many variations and consequences of the original theorem have been proved subsequently, and we provide an account of this branch of operator theory.

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Section: Infinite Systems or Coupled Ordinary Differential Equationsmentioning

confidence: 99%

Some developments around the Katznelson–Tzafriri theorem

Batty

Seifert

2022

Acta Sci. Math. (Szeged)

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“…This time, however, the estimates are less straightforward and moreover [11,Theorem 5.1] involves a logarithmic term in the estimate for the rate of convergence which was conjectured in [11, Remark 5.2(a)] to be unnecessary. It is shown in our recent paper [10] how the argument outlined in the proof of Theorem 3.1 above can be extended to the more general setting of [11], thus in particular removing the logarithm in the platoon model.…”

Section: Further Extensionsmentioning

confidence: 99%

Asymptotic behaviour in the robot rendezvous problem

Paunonen

Seifert²

2017

Automatica

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Abstract. This paper presents a natural extension of the results obtained by Feintuch and Francis in [5,6] concerning the so-called robot rendezvous problem. In particular, we revisit a known necessary and sufficient condition for convergence of the solution in terms of Cesàro convergence of the translates S k x0, k ≥ 0, of the sequence x0 of initial positions under the right-shift operator S, thus shedding new light on questions left open in [5,6]. We then present a new proof showing that a certain stronger ergodic condition on x0 ensures that the corresponding solution converges to its limit at the optimal rate O(t −1/2 ) as t → ∞. After considering a natural two-sided variant of the robot rendezvous problem already studied in [5] and in particular proving a new quantified result in this case, we conclude by relating the robot rendezvous problem to a more realistic model of vehicle platoons.

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“…Both authors were at least partially motivated by concrete applications, which in Nevanlinna's case come from the theory of iterative methods, in Dungey's from the theory of random walks and Markov processes which had inspired the original discovery of Theorem 1.1; see [12,13,15] and also [9,10]. Other applications arise for instance in the theory of alternating projections [1,2], the study of certain periodic evolution equations [28] and even in investigations of infinite systems of coupled ordinary differential equations [27]. Surveys touching on various aspects of the Katznelson-Tzafriri theorem and its quantified versions may be found in [8,21].…”

Section: Introductionmentioning

confidence: 99%

Optimal rates of decay in the Katznelson-Tzafriri theorem for operators on Hilbert spaces

Ng,

Seifert

2020

Preprint

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The Katznelson-Tzafriri theorem is a central result in the asymptotic theory of discrete operator semigroups. It states that for a power-bounded operator T on a Banach space we have ||T n (I −T ) → 0 if and only if σ(T ) ∩ T ⊆ {1}. The main result of the present paper gives a sharp estimate for the rate at which this decay occurs for operators on Hilbert space, assuming the growth of the resolvent norms R(e iθ , T ) as |θ| → 0 satisfies a mild regularity condition. This significantly extends an earlier result by the second author, which covered the important case of polynomial resolvent growth. We further show that, under a natural additional assumption, our condition on the resolvent growth is not only sufficient but also necessary for the conclusion of our main result to hold. By considering a suitable class of Toeplitz operators we show that our theory has natural applications even beyond the setting of normal operators, for which we in addition obtain a more general result.

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Asymptotic Behaviour of Coupled Systems in Discrete and Continuous Time

Cited by 5 publications

References 15 publications

Some developments around the Katznelson–Tzafriri theorem

Some developments around the Katznelson–Tzafriri theorem

Asymptotic behaviour in the robot rendezvous problem

Optimal rates of decay in the Katznelson-Tzafriri theorem for operators on Hilbert spaces

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