Semigroups Associated with Dissipative Systems

1. Auflage 1999
ISBN: 978-0-8493-0615-0
Verlag: Chapman and Hall/CRC

Buch, Englisch, Band 398, 224 Seiten, Format (B × H): 159 mm x 243 mm, Gewicht: 342 g

Reihe: Chapman & Hall/CRC Research Notes in Mathematics Series

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.

Zielgruppe

Researchers and graduate students in functional analysis, partial differential equations, control theory, solid mechanics, and visoelasticity

Autoren/Hrsg.

Liu, Zhuangyi

Liu, Z

Zheng, Songmu

Fachgebiete

• Technische Wissenschaften Maschinenbau | Werkstoffkunde Technische Mechanik | Werkstoffkunde Kontinuumsmechanik
• Mathematik | Informatik Mathematik Algebra Algebraische Strukturen, Gruppentheorie
• Mathematik | Informatik Mathematik Mathematische Analysis Funktionalanalysis
• Mathematik | Informatik Mathematik Mathematische Analysis Harmonische Analysis, Fourier-Mathematik
• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen

Inhaltsverzeichnis

PreliminariesSome DefinitionsC0-Semigroup Generated by Dissipative OperatorExponential Stability and AnalyticityThe Sobolev Spaces and Elliptic Boundary Value ProblemsLinear Thermoelastic SystemsThe Setting of Problems for the One-Dimensional Thermoelastic SystemThe Exponential Stability for the Dirichlet Boundary Conditions at Both EndsThe Exponential Stability for the Stress-Free Boundary Conditions at Both EndsThe Exponential Stability for the Stress-Free Boundary Conditions at One EndThe Thermoelastic Kirchhoff Plate EquationsLinear Viscoelastic SystemLinear Viscoelastic SystemWave Equation with Locally Distributed DampingLinear Viscoelastic System with MemoryThe Linear Viscoelastic Kirchoff Plate with MemoryLinear Thermoviscoelastic SystemsLinear One-Dimensional Thermoviscoelastic SystemLinear Three-Dimensional Thermoviscoelastic System with MemoryElastic Systems with Shear DampingShear Diffusion EquationsLaminated Beam with Shear DampingLinear Elastic Systems with Boundary DampingSecond-Order Hyperbolic EquationEuler-Bernoulli Beam EquationUniformly Stable ApproximationsMain TheoremApproximations of the Thermoelastic SystemApproximation of the Viscoelastic SystemBibliography