A Note on Exponential Stability for Evolutionary Equations.

Trostorff, Sascha

doi:10.1002/pamm.201410472

Cited by 7 publications

(7 citation statements)

References 76 publications

(126 reference statements)

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“…Then,

C_{MathClass-rel\infty MathClass-punc, c} (double-struckR MathClass-punc; H)

is dense in

L_{ν MathClass-rel'}^{2} (double-struckR MathClass-punc; H)

for every

ν MathClass-rel' ≧ ν

. Moreover, Cauchy's integral theorem shows that

{scriptL}_{ν MathClass-rel'}^{MathClass-bin*} M ((), \frac{1}{im MathClass-bin+ ν MathClass-rel'}) {scriptL}_{ν MathClass-rel'} φ MathClass-rel= {scriptL}_{ν}^{MathClass-bin*} M ((), \frac{1}{im MathClass-bin+ ν}) {scriptL}_{ν} φ

for every

ν MathClass-rel' ≧ ν

and

φ MathClass-rel\in C_{MathClass-rel\infty MathClass-punc, c} (double-struckR MathClass-punc; H)

; see also , Lemma 3.6 for more details. Moreover, the operator norm of

M (\partial_{0}^{MathClass-bin- 1})

L_{ν MathClass-rel'}^{2} (double-struckR MathClass-punc; H)

is easily estimated by sup z ∈ B ( r , r ) ∥ M ( z ) ∥ L ( H ) . Example Let H be a Hilbert space and let

L_{s}^{MathClass-rel\infty} (double-struckR MathClass-punc; L (H))

be the space of bounded strongly measurable functions from

double-struckR

to L ( H ).…”

Section: Preliminariesmentioning

confidence: 98%

On non‐autonomous integro‐differential‐algebraic evolutionary problems

Waurick

2014

Math Methods in App Sciences

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In this article, we show that a technique for showing well‐posedness results for evolutionary equations in the sense of Picard and McGhee [Picard, McGhee, Partial Differential Equations: A unified Hilbert Space Approach, DeGruyter, Berlin, 2011] established in [Picard, Trostorff, Wehowski, Waurick, On non‐autonomous evolutionary problems. J. Evol. Equ. 13:751‐776, 2013] applies to a broader class of non‐autonomous integro‐differential‐algebraic equations. Using the concept of evolutionary mappings, we prove that the respective solution operators do not depend on certain parameters describing the underlying spaces in which the well‐posedness results are established. Copyright © 2014 John Wiley & Sons, Ltd.

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“…Then,

C_{MathClass-rel\infty MathClass-punc, c} (double-struckR MathClass-punc; H)

is dense in

L_{ν MathClass-rel'}^{2} (double-struckR MathClass-punc; H)

for every

ν MathClass-rel' ≧ ν

. Moreover, Cauchy's integral theorem shows that

{scriptL}_{ν MathClass-rel'}^{MathClass-bin*} M ((), \frac{1}{im MathClass-bin+ ν MathClass-rel'}) {scriptL}_{ν MathClass-rel'} φ MathClass-rel= {scriptL}_{ν}^{MathClass-bin*} M ((), \frac{1}{im MathClass-bin+ ν}) {scriptL}_{ν} φ

for every

ν MathClass-rel' ≧ ν

and

φ MathClass-rel\in C_{MathClass-rel\infty MathClass-punc, c} (double-struckR MathClass-punc; H)

; see also , Lemma 3.6 for more details. Moreover, the operator norm of

M (\partial_{0}^{MathClass-bin- 1})

L_{ν MathClass-rel'}^{2} (double-struckR MathClass-punc; H)

is easily estimated by sup z ∈ B ( r , r ) ∥ M ( z ) ∥ L ( H ) . Example Let H be a Hilbert space and let

L_{s}^{MathClass-rel\infty} (double-struckR MathClass-punc; L (H))

be the space of bounded strongly measurable functions from

double-struckR

to L ( H ).…”

Section: Preliminariesmentioning

confidence: 98%

On non‐autonomous integro‐differential‐algebraic evolutionary problems

Waurick

2014

Math Methods in App Sciences

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“…To find the right conditions on k to ensure (b) is more delicate and will be postponed to future research. In case of a kernel k(t, s) = k(t−s), this was done in [29,30] even for operator-valued kernels.…”

Section: The Main Resultsmentioning

confidence: 99%

Well-posedness for a general class of differential inclusions

Trostorff

2020

Journal of Differential Equations

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We consider an abstract class of differential inclusions, which covers differentialalgebraic and non-autonomous problems as well as problems with delay. Under weak assumptions on the operators involved, we prove the well-posedness of those differential inclusions in a pure Hilbert space setting. Moreover, we study the causality of the associated solution operator. The theory is illustrated by an application to a semistatic quasilinear variant of Maxwell's equations.

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“…We are now ready to introduce the notion of an exponentially stable evolutionary problem as it was defined in [20,21].…”

Section: Well-posedness and Exponential Stabilitymentioning

confidence: 99%

“…Hence, we need to introduce a more general notion of exponential stability for that class of problems. This was done by the author in [20,21] (see also Subsection 2.2 in this article), where also sufficient conditions on the material law M to obtain exponential stability were derived.…”

Section: Introductionmentioning

confidence: 99%