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

Only in the last 15 years or so has the notion of semi-uniform stability, which lies between exponential stability and strong stability, become part of the asymptotic theory of C 0 -semigroups. It now lies at the very heart of modern semigroup theory. After briefly reviewing the notions of exponential and strong stability, we present an overview of some of the best known (and often optimal) abstract results on semi-uniform stability. We go on to indicate briefly how these results can be applied to obtain (sometimes optimal) rates of energy decay for certain damped second-order Cauchy problems. This article is part of the theme issue ‘Semigroup applications everywhere’.

Download Full-text

Uniform stability for a type III thermoelastic laminated beam with structural damping

10.22541/au.164190249.91339348/v1 ◽

2022 ◽

Author(s):

Fayssal Djellali

Keyword(s):

Heat Flux ◽

Exponential Stability ◽

Semigroup Theory ◽

Stability Result ◽

Uniform Stability ◽

Well Posedness ◽

Structural Damping ◽

Laminated Beam ◽

Wave Speeds ◽

Lack Of Exponential Stability

In this work, we consider a thermoelastic laminated beam with structural damping, where the heat flux is given by Green and Naghdi theories. We establish the well-posedness of the system using semigroup theory. Moreover, under the condition of equal wave speeds, we prove an exponential stability result for the considered system. In the case of lack of exponential stability we show that the solution decays polynomially.

Download Full-text

Semigroups for dynamical processes on metric graphs

Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rsta.2019.0619 ◽

2020 ◽

Vol 378 (2185) ◽

pp. 20190619

Author(s):

Marjeta Kramar Fijavž ◽

Aleksandra Puchalska

Keyword(s):

Dynamical Systems ◽

Second Order ◽

Well Posedness ◽

Biological Models ◽

Theme Issue ◽

Metric Graphs ◽

Operator Semigroups ◽

Dynamical Processes

We present the operator semigroups approach to the first- and second-order dynamical systems taking place on metric graphs. We briefly survey the existing results and focus on the well-posedness of the problems with standard vertex conditions. Finally, we show two applications to biological models. This article is part of the theme issue ‘Semigroup applications everywhere’.

Download Full-text

Existence and exponential stability in the pth moment for impulsive neutral stochastic integro-differential equations driven by mixed fractional Brownian motion

Journal of Inequalities and Applications ◽

10.1186/s13660-019-2213-5 ◽

2019 ◽

Vol 2019 (1) ◽

Cited By ~ 1

Author(s):

Xia Zhou ◽

Dongpeng Zhou ◽

Shouming Zhong

Keyword(s):

Brownian Motion ◽

Differential Equations ◽

Exponential Stability ◽

Fractional Brownian Motion ◽

Mild Solutions ◽

Semigroup Theory ◽

Sufficient Conditions ◽

Fractional Power ◽

Banach Fixed Point Theorem ◽

Mixed Fractional Brownian Motion

Abstract This paper consider the existence, uniqueness and exponential stability in the pth moment of mild solution for impulsive neutral stochastic integro-differential equations driven simultaneously by fractional Brownian motion and by standard Brownian motion. Based on semigroup theory, the sufficient conditions to ensure the existence and uniqueness of mild solutions are obtained in terms of fractional power of operators and Banach fixed point theorem. Moreover, the pth moment exponential stability conditions of the equation are obtained by means of an impulsive integral inequality. Finally, an example is presented to illustrate the effectiveness of the obtained results.

Download Full-text

Second order sensitivity analysis and asymptotic theory of parametrized nonlinear programs

Mathematical Programming ◽

10.1007/bf01584378 ◽

1985 ◽

Vol 33 (3) ◽

pp. 280-299 ◽

Cited By ~ 45

Author(s):

Alexander Shapiro

Keyword(s):

Sensitivity Analysis ◽

Asymptotic Theory ◽

Second Order ◽

Nonlinear Programs

Download Full-text

Asymptotic Theory of Low-Degree Stellar Acoustic Oscillations

International Astronomical Union Colloquium ◽

10.1017/s0252921100018364 ◽

1993 ◽

Vol 137 ◽

pp. 535-537 ◽

Cited By ~ 1

Author(s):

I.W. Roxburgh ◽

S.V. Vorontsov

Keyword(s):

Asymptotic Theory ◽

Second Order ◽

Solar Model ◽

Acoustic Oscillations ◽

Stellar Envelope ◽

Asymptotic Description ◽

Low Degree

AbstractWe extend the second-order asymptotic description developed by Tassoul (1980, 1990) to the forth order, taking into account both gravity perturbations and realistic (non-polytropic) structure of the stellar envelope. We examine the accuracy of the asymptotic description by the direct computations for a solar model.

Download Full-text

Abstract Cauchy problems for second order linear differential equations in a Banach space

Hiroshima Mathematical Journal ◽

10.32917/hmj/1206129964 ◽

1987 ◽

Vol 17 (3) ◽

pp. 591-612 ◽

Cited By ~ 4

Author(s):

Tosiharu Takenaka ◽

Noboru Okazawa

Keyword(s):

Banach Space ◽

Differential Equations ◽

Second Order ◽

Linear Differential Equations ◽

Cauchy Problems ◽

Order Linear ◽

Linear Differential ◽

Abstract Cauchy Problems

Download Full-text

The liouville-green asymptotic theory for second-order differential equations: A new approach and some extensions

Lecture Notes in Mathematics - Ordinary Differential Equations and Operators ◽

10.1007/bfb0076795 ◽

1983 ◽

pp. 110-122 ◽

Cited By ~ 1

Author(s):

M. S. P. Eastham

Keyword(s):

Differential Equations ◽

Asymptotic Theory ◽

Second Order ◽

New Approach ◽

Second Order Differential Equations

Download Full-text

Exponential Stability Estimate of Fully Nonlinear Aceive Equation by Boundary Control

Key Engineering Materials ◽

10.4028/www.scientific.net/kem.467-469.1078 ◽

2011 ◽

Vol 467-469 ◽

pp. 1078-1083

Author(s):

Dian Chen Lu ◽

Ruo Yu Zhu

Keyword(s):

Fixed Point ◽

Exponential Stability ◽

Dispersion Equation ◽

Boundary Control ◽

Existence And Uniqueness ◽

Stability Estimate ◽

Banach Fixed Point Theorem ◽

Operator Semigroups ◽

Fully Nonlinear ◽

Well Posed

The well-posed problem for the fully nonlinear Aceive diffusion and dispersion equation on the domain [0, 1] is investigated by using boundary control. The existence and uniqueness of the solutions with the help of the Banach fixed point theorem and the theory of operator semigroups are verified. By using some inequalities and integration by parts, the exponential stability of the fully nonlinear Aceive diffusion and dispersion equation with the designed boundary feedback is also proved.

Download Full-text