This monograph examines the stability of various coupled systems with local Kelvin-Voigt damping. The development of this area is thoroughly reviewed along with the authors’ contributions. New results are featured on the fundamental properties of solutions of linear transmission evolution PDEs involving Kelvin-Voigt damping, with special emphasis on the asymptotic behavior of these solutions. The vibrations of transmission problems are highlighted as well, making this a valuable resource for those studying this active area of research.
The book begins with a brief description of the abstract theory of linear evolution equations with a particular focus on semigroup theory. Different types of stability are also introduced along with their connection to resolvent estimates. After this foundation is established, different models are presented for uni-dimensional and multi-dimensional linear transmission evolution partial differential equations with Kelvin-Voigt damping. Stabilization of Kelvin-Voigt Damped Systems will be a useful reference for researchers in mechanics, particularly those interested in the study of control theory of PDEs.
Author(s): Kaïs Ammari, Fathi Hassine
Series: Advances in Mechanics and Mathematics, 47
Publisher: Birkhäuser
Year: 2022
Language: English
Pages: 155
City: Cham
Preface
Contents
1 Preliminaries
1.1 Semigroups of Bounded Linear Operators
1.1.1 Basic Properties
1.2 Stability
1.2.1 Strong Stability
1.2.2 Exponential Stability
1.2.3 Polynomial Stability
1.2.4 Logarithmic Stability
2 Stability of Elastic Transmission Systems with a Local Kelvin–Voigt Damping
2.1 Introduction
2.2 Transversal Motion
2.3 Longitudinal Motion
3 Stabilization for the Wave Equation with Singular Kelvin–Voigt Damping
3.1 Introduction and Main Results
3.2 Well-posedness and Strong Stability
3.3 Carleman Estimate
3.4 Stabilization Result
4 Logarithmic Stabilization of the Euler–Bernoulli Transmission Plate Equation with Locally Distributed Kelvin–Voigt Damping
4.1 Introduction
4.2 Model and Statement of Results
4.3 Existence and Uniqueness
4.4 Carleman Estimate and Construction of the Weight Functions
4.4.1 Carleman Estimate
4.4.2 Construction of the Weight Functions
4.5 Resolvent Estimate
5 Energy Decay Estimates of Elastic Transmission Wave/Beam Systems with a Local Kelvin–Voigt Damping
5.1 Introduction and Motivation
5.2 Preliminary and Main Results
5.3 Damping Arising from the Transversal Motion
5.4 Damping Arising from the Longitudinal Motion
5.5 Non-exponential Stability
5.5.1 Case When the Damping Arising from the Wave Equation
5.5.2 Case When the Damping Arising from the Beam Equation
6 Asymptotic Behavior of the Transmission Euler–Bernoulli Plate and Wave Equation with a Localized Kelvin–Voigt Damping
6.1 Introduction
6.1.1 The Method of Analysis
6.2 Existence and Uniqueness
6.3 Carleman Estimate Near the Surface
6.4 Resolvent Estimate
7 Conclusion and Perspectives
References
