Semigroups Associated with Dissipative Systems
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Summary
Motivated by applications to control theory and to the theory of partial differential equations (PDE's), the authors examine the exponential stability and analyticity of C0-semigroups associated with various dissipative systems. They present a unique, systematic approach in which they prove exponential stability by combining a theory from semigroup theory with partial differential equation techniques, and use an analogous theorem with PDE techniques to prove analyticity. The result is a powerful but simple tool useful in determining whether these properties will preserve for a given dissipative system.
The authors show that the exponential stability is preserved for all the mechanical systems considered in this book-linear, one-dimensional thermoelastic, viscoelastic and thermoviscoelastic systems, plus systems with shear or friction damping. However, readers also learn that this property does not hold true for linear three-dimensional systems without making assumptions on the domain and initial data, and that analyticity is a more sensitive property, not preserved even for some of the systems addressed in this study.
Table of Contents
Preliminaries
Some Definitions
C0-Semigroup Generated by Dissipative Operator
Exponential Stability and Analyticity
The Sobolev Spaces and Elliptic Boundary Value Problems
Linear Thermoelastic Systems
The Setting of Problems for the One-Dimensional Thermoelastic System
The Exponential Stability for the Dirichlet Boundary Conditions at Both Ends
The Exponential Stability for the Stress-Free Boundary Conditions at Both Ends
The Exponential Stability for the Stress-Free Boundary Conditions at One End
The Thermoelastic Kirchhoff Plate Equations
Linear Viscoelastic System
Linear Viscoelastic System
Wave Equation with Locally Distributed Damping
Linear Viscoelastic System with Memory
The Linear Viscoelastic Kirchoff Plate with Memory
Linear Thermoviscoelastic Systems
Linear One-Dimensional Thermoviscoelastic System
Linear Three-Dimensional Thermoviscoelastic System with Memory
Elastic Systems with Shear Damping
Shear Diffusion Equations
Laminated Beam with Shear Damping
Linear Elastic Systems with Boundary Damping
Second-Order Hyperbolic Equation
Euler-Bernoulli Beam Equation
Uniformly Stable Approximations
Main Theorem
Approximations of the Thermoelastic System
Approximation of the Viscoelastic System
Bibliography
Editorial Reviews
"The book is clearly written and recommended especially for researchers and graduate students in the field of functional analysis, partial differential equations and control theory."
-Reinhard Racke, Mathematical Reviews
