Semi-uniform stability of operator semigroups and energy decay of damped waves

Chill, Ralph; Seifert, David; Tomilov, Yuri

doi:10.1098/rsta.2019.0614

Cited by 22 publications

(13 citation statements)

References 137 publications

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“…The proof of Theorem 3.3 is divided into several steps. In the next subsection, we give the equivalence between the stability of the sampled-data system (5) and that of the discretized system. Section 4 is devoted to resolvent conditions for the stability of discrete semigroups on Hilbert spaces.…”

Section: Resultsmentioning

confidence: 99%

“…Various results on polynomial stability, and more generally semi-uniform stability, have been obtained such as characterizations of decay rates by resolvent estimates on the imaginary axis iR [2-4, 15, 30] and robustness to perturbations [20][21][22][23]27]. We also refer to [5] for an overview of semi-uniform stability. A discrete version of semi-uniform stability has been investigated in the context of the quantified Katznelson-Tzafriri theorem [6,18,31,32] and the Cayley transform of a generator [37].…”

Section: Introductionmentioning

confidence: 99%

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Robustness of Polynomial Stability with respect to Sampling

Wakaiki¹

2022

Preprint

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In this paper, we provide a partially affirmative answer to the following question on robustness of stability with respect to sampling: "Suppose that a continuous-time state-feedback controller achieves the polynomial stability of the infinite-dimensional linear system. We apply an idealized sampler and a zero-order hold to a feedback loop around the controller. Then, is the sampled-data system strongly stable for all sufficiently small sampling periods? Furthermore, is the polynomial decay of the continuous-time system inherited to the sampled-data system under sufficiently fast sampling?" The generator of the open-loop system is assumed to be a Riesz-spectral operator whose eigenvalues are not on the imaginary axis but may approach it asymptotically. We provide conditions for strong stability to be preserved under fast sampling. Moreover, we estimate the decay rate of the state of the sampled-data system with a smooth initial state and a sufficiently small sampling period.

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Robustness of Polynomial Stability with respect to Sampling

Wakaiki¹

2022

Preprint

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“…The already mentioned article [2] is an overview article on semi-uniform stability. We remark that this notion is sometimes called differently, e.g.…”

Section: Stability Resultsmentioning

confidence: 99%

Stability of the multidimensional wave equation in port-Hamiltonian modelling

Jacob,

Skrepek

2021

Preprint

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We investigate the stability of the wave equation with spatial dependent coefficients on a bounded multidimensional domain. The system is stabilized via a scattering passive feedback law. We formulate the wave equation in a port-Hamiltonian fashion and show that the system is semi-uniform stable, which is a stability concept between exponential stability and strong stability. Hence, this also implies strong stability of the system. In particular, classical solutions are uniformly stable. This will be achieved by showing that the spectrum of the port-Hamiltonian operator is contained in the left half plane C − and the port-Hamiltonian operator generates a contraction semigroup. Moreover, we show that the spectrum consists of eigenvalues only and the port-Hamiltonian operator has a compact resolvent.

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“…This work has been used extensively to obtain rates of decay for damped wave equations (see e.g. the references in [8,19]).…”

Section: Discussion Of (P2)mentioning

confidence: 99%