Functional Analysis and Linear Control Theory
Published on 5. December 1980
160 pages
978-0-08-095998-6 (ISBN)
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Functional Analysis and Linear Control Theory
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J. R. Leigh
Functional Analysis and Linear Control Theory
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10/1997
Academic Press
€82.94
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Content
- Front Cover
- Functional Analysis and Linear Control Theory
- Copyright Page
- Contents
- Preface
- Chapter 1. Preliminaries
- 1.1 Control Theory
- 1.2 Set Theory
- 1.3 Linear Space (Vector Space)
- 1.4 Linear Independence
- 1.5 Maximum and Supremum
- 1.6 Metric and Norm
- 1.7 Sequences and Limit Concepts
- 1.8 Convex Sets
- 1.9 Intervals on a Line
- 1.10 The K Cube
- 1.11 Product Sets and Product Spaces
- 1.12 Direct Sum
- 1.13 Functions and Mappings
- 1.14 Exercises
- Chapter 2. Basic Concepts
- 2.1 Topological Concepts
- 2.2 Compactness
- 2.3 Convergence
- 2.4 Measure Theory
- 2.5 Euclidean Spaces
- 2.6 Sequence Spaces
- 2.7 The Lebesgue Integral
- 2.8 Spaces of Lebesgue Integrable Functions (Lp Spaces)
- 2.9 Inclusion Relations between Sequence Spaces
- 2.10 Inclusion Relations between Function Spaces on a Finite Interval
- 2.11 The Hierarchy of Spaces
- 2.12 Linear Functionals
- 2.13 The Dual Space
- 2.14 The Space of all Bounded Linear Mappings
- 2.15 Exercises
- Chapter 3. Inner Product Spaces and Some of their Properties
- 3.1 Inner Product
- 3.2 Orthogonality
- 3.3 Hilbert Space
- 3.4 The Parallelogram Law
- 3.5 Theorems
- 3.6. Exercises
- Chapter 4. Some Major Theorems of Functional Analysis
- 4.1 Introduction
- 4.2 The Hahn-Banach Theorem and its Geometric Equivalent
- 4.3 Other Theorems Related to Mappings
- 4.4 Hölder's Inequality
- 4.5 Norms on Product Spaces
- 4.6 Exercises
- Chapter 5. Linear Mappings and Reflexive Spaces
- 5.1 Introduction
- 5.2 Mappings of Finite Rank
- 5.3 Mappings of Finite Rank on a Hilbert Space
- 5.4 Reflexive Spaces
- 5.5 Rotund Spaces
- 5.6 Smooth Spaces
- 5.7 Uniform Convexity
- 5.8 Convergence in Norm (Strong Convergence)
- 5.9 Weak Convergence
- 5.10 Weak Compactness
- 5.11 Weak* Convergence and Weak* Compactness
- 5.12 Weak Topologies
- 5.13 Failure of Compactness in Infinite Dimensional Spaces
- 5.14 Convergence of Operators
- 5.15 Weak, Strong and Uniform Continuity
- 5.16 Exercises
- Chapter 6. Axiomatic Representation of Systems
- 6.1 Introduction
- 6.2 The Axioms
- 6.3 Relation between the Axiomatic Representation and the Representation as a Finite Set of Differential Equations
- 6.4 Visualization of the Concepts of this Chapter
- 6.5 System Realization
- 6.6 The Transition Matrix and Some of its Properties
- 6.7 Calculation of the Transition Matrix for Time Invariant Systems
- 6.8 Exercises
- Chapter 7. Stability, Controllability and Observability
- 7.1 Introduction
- 7.2 Stability
- 7.3 Controllability and Observability
- 7.4 Exercises
- Chapter 8. Minimum Norm Control
- 8.1 Introduction
- 8.2 Minimum Norm Problems: Literature
- 8.3 Minimum Norm Problems: Outline of the Approach
- 8.4 Minimum Norm Problem in Hilbert Space: Definition
- 8.5 Minimum Norm Problems in Banach Space
- 8.6 More General Optimization Problems
- 8.7 Minimum Norm Control: Characterization, a Simple Example
- 8.8 Development of Numerical Methods for the Calculation of Minimum Norm Controls
- 8.9 Exercises
- Chapter 9. Minimum Time Control
- 9.1 Preliminaries and Problem Description
- 9.2 The Attainable Set
- 9.3 Existence of a Minimum Time Control
- 9.4 Uniqueness
- 9.5 Characterization
- 9.6 The Pontryagin Maximum Principle
- 9.7 Time Optimal Control
- 9.8 Exercises
- Chapter 10. Distributed Systems
- 10.1 Introduction
- 10.2 Further Theorems From Functional Analysis
- 10.3 Axiomatic Description
- 10.4 Representation of Distributed Systems
- 10.5 Characterization of The Solution of the Equation x =Ax + Bu
- 10.6 Stability
- 10.7 Controllability
- 10.8 Minimum Norm Control
- 10.9 Time-Optimal Control
- 10.10 Optimal Control of a Distributed System: An Example
- 10.11 Approximate Numerical Solution
- 10.12 Exercises
- Glossary of Symbols
- References and Further Reading
- Subject Index
