Non-uniform Stability of Damped Contraction Semigroups

Chill, Ralph; Paunonen, Lassi; Seifert, David; Stahn, Reinhard; Tomilov, Yuri

doi:10.48550/arxiv.1911.04804

Cited by 4 publications

(6 citation statements)

References 60 publications

(120 reference statements)

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“…Our assumptions together with the results in [15] and [11,Prop. 3.9] also imply that the condition (2.5) is satisfied for some functions η : R → (0, η 0 ] and δ : R → (0, δ 0 ] satisfying η(s) −2 δ(s) −2 ≤ M 0 (1 + s 2 ) for all s ∈ R. Because of this, Theorem 2.3 could in principle be used to derive numerical values for κ > 0 for particular damping functions d(•, •).…”

Section: Perturbations Of Damped 2-d Wave Equationsmentioning

confidence: 90%

“…The following theorem introduces a concrete bound κ > 0 for the norms of the perturbations in Theorem 2.2 for A = A 0 − DD * with a skew-adjoint operator A 0 in the important special case β, γ ≥ 0 are chosen so that β + γ = ⌈α⌉ (here ⌈α⌉ ∈ N denotes the ceiling of α > 0). The first part of the result is a special case of [11,Thm. 3.5] with a proof that has been modified in a trivial manner to yield an explicit constant M R > 0.…”

Section: Definition 21 ([8]mentioning

confidence: 96%

“…The polynomial stability of (1.1) means that there exist constants α, M > 0 such that for all initial conditions w 0 ∈ dom L and w 1 ∈ dom (−L) 1/2 the solutions of (1.1) satisfy [8] (−L) 1/2 w(t) 2 + w t (t) 2 ≤ M t 2/α Lw 0 2 + (−L) 1/2 w 1 2 , t > 0. (1.2) Polynomial stability has been investigated in detail in the literature for damped wave equations on multi-dimensional domains [17,9,3], coupled partial differential equations [27,13,4,24], as well as abstract damped second order systems of the form (1.1) [2,18,1,11].…”

Section: Introductionmentioning

confidence: 99%

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Robustness of polynomial stability of damped wave equations

Baidiuk¹,

Paunonen²

2020

Preprint

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In this paper we present new results on the preservation of polynomial stability of damped wave equations under addition of perturbing terms. We in particular introduce sufficient conditions for the stability of perturbed two-dimensional wave equations on rectangular domains, a one-dimensional weakly damped Webster's equation, and a wave equation with an acoustic boundary condition. In the case of Webster's equation, we use our results to compute explicit numerical bounds that guarantee the polynomial stability of the perturbed equation.

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Section: Perturbations Of Damped 2-d Wave Equationsmentioning

confidence: 90%

Section: Definition 21 ([8]mentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

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Robustness of polynomial stability of damped wave equations

Baidiuk¹,

Paunonen²

2020

Preprint

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“…2.6] (see also [32,Rem. 2.7]), which is a discrete analogue of [6,Prop. 5.4] and proves that decay strictly faster than m −1 (cn) for all c > 0 as n → ∞ is impossible provided R(e iθ , T ) does not grow strictly more slowly than m(|θ|) as θ → 0.…”

Section: An Optimal Upper Boundmentioning

confidence: 99%

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

Ng,

Seifert

2020

Preprint

Self Cite

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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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“…We have thus obtained that the pair (A, B) satisfies the assumptions of Theorem 1.1 in [21] with δ given by (6.14) and…”

mentioning

confidence: 91%

Stabilizability properties of a linearized water waves system

Tucsnak

2020

Systems & Control Letters

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We consider the strong stabilization of small amplitude gravity water waves in a two dimensional rectangular domain. The control acts on one lateral boundary, by imposing the horizontal acceleration of the water along that boundary, as a multiple of a scalar input function u, times a given function h of the height along the active boundary. The state z of the system consists of two functions: the water level ζ along the top boundary, and its time derivativeζ. We prove that for suitable functions h, there exists a bounded feedback functional F such that the feedback u = F z renders the closed-loop system strongly stable. Moreover, for initial states in the domain of the semigroup generator, the norm of the solution decays like (1+t) − 1 6 . Our approach uses a detailed analysis of the partial Dirichlet to Neumann and Neumann to Neumann operators associated to certain edges of the rectangular domain, as well as recent abstract non-uniform stabilization results by Chill, Paunonen, Seifert, Stahn and Tomilov (2019).If H is a Hilbert space, D(A 0 ) is a subspace of H and A 0 : D(A 0 ) → H is a linear operator, then A 0 is called strictly positive if A 0 is self-adjoint and there exists m 0 > 0 such that

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Non-uniform Stability of Damped Contraction Semigroups

Cited by 4 publications

References 60 publications

Robustness of polynomial stability of damped wave equations

Robustness of polynomial stability of damped wave equations

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

Stabilizability properties of a linearized water waves system

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