1st Edition
Modeling and Control in Vibrational and Structural Dynamics A Differential Geometric Approach
Modeling and Control in Vibrational and Structural Dynamics: A Differential Geometric Approach describes the control behavior of mechanical objects, such as wave equations, plates, and shells. It shows how the differential geometric approach is used when the coefficients of partial differential equations (PDEs) are variable in space (waves/plates), when the PDEs themselves are defined on curved surfaces (shells), and when the systems have quasilinear principal parts.
To make the book self-contained, the author starts with the necessary background on Riemannian geometry. He then describes differential geometric energy methods that are generalizations of the classical energy methods of the 1980s. He illustrates how a basic computational technique can enable multiplier schemes for controls and provide mathematical models for shells in the form of free coordinates. The author also examines the quasilinearity of models for nonlinear materials, the dependence of controllability/stabilization on variable coefficients and equilibria, and the use of curvature theory to check assumptions.
With numerous examples and exercises throughout, this book presents a complete and up-to-date account of many important advances in the modeling and control of vibrational and structural dynamics.
Preliminaries from Differential Geometry
Linear Connections, Differential of Tensor Fields, and Curvature
Distance Functions
A Basic Computational Technique
Sobolev Spaces of Tensor Field and Some Basic Differential Operators
Control of the Wave Equation with Variable Coefficients in Space
How to Understand Riemannian Geometry as a Necessary Tool for Control of the Wave Equation with Variable Coefficients
Geometric Multiplier Identities
Escape Vector Fields and Escape Regions for Metrics
Exact Controllability. Dirichlet/Neumann Action
Smooth Controls
A Counterexample without Exact Controllability
Stabilization
Transmission Stabilization
Control of the Plate with Variable Coefficients in Space
Multiplier Identities
Escape Vector Fields for the Plate
Exact Controllability from Boundary
Controllability for Transmission of Plate
Stabilization from Boundary for the Plate with a Curved Middle Surface
Linear Shallow Shells: Modeling and Control
Equations in Equilibrium. Green’s Formulas
Ellipticity of the Strain Energy of Shallow Shells
Equations of Motion
Multiplier Identities
Escape Vector Field and Escape Region for the Shallow Shell
Observability Inequalities. Exact Controllability
Exact Controllability for Transmission
Stabilization by Linear Boundary Feedbacks
Stabilization by Nonlinear Boundary Feedbacks
Naghdi’s Shells: Modeling and Control
Equations of Equilibrium. Green’s Formulas. Ellipticity of the Strain Energy. Equations of Motion
Observability Estimates from Boundary
Stabilization by Boundary Feedback
Stabilization of Transmission
Koiter’s Shells: Modeling and Controllability
Equations of Equilibria. Equations of Motion
Uniqueness for the Koiter Shell
Multiplier Identities
Observability Estimates from Boundary
Control of the Quasilinear Wave Equation in Higher Dimensions
Boundary Traces and Energy Estimates
Locally and Globally Boundary Exact Controllability
Boundary Feedback Stabilization
Structure of Control Regions for Internal Feedbacks
References
Bibliography
Index
Notes and References appear at the end of each chapter.
Biography
Peng-Fei Yao is a professor in the Key Laboratory of Systems and Control in the Chinese Academy of Sciences. His research interests include control and modeling of vibrational mechanics, the scattering problem of vibrational systems, global and blow-up solutions, and nonlinear elasticity.
