Functional Analysis and Linear Control Theory

✍ Scribed by J. R. Leigh

• Publisher: Academic Press
• Year: 1981
• Tongue: English
• Leaves: 171
• Edition: 1
• Category: Library

✦ Synopsis

Functional analysis provides a concise conceptual framework for linear control theory. This self-contained text, geared toward engineering students, demonstrates the subject's unity. A wide range of powerful theorems illustrates inner product spaces; instability, controllability, and observability; minimum norm and time control; and distributed systems. 1980 edition.

✦ Table of Contents

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