Introduction to the Mathematical Theory of Control Processes

Name: Introduction to the Mathematical Theory of Control Processes | Linear Equations and Quadratic Criteria
Brand: Academic Press
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Linear Equations and Quadratic Criteria

Richard Bellman(Author)

Academic Press

Published on 3. June 2016

264 pages

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978-1-4832-2286-8 (ISBN)

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Front Cover
Linear Equations and Quadratic Criteria
Copyright Page
Table of Contents
INTRODUCTION
Chapter 1. WHAT IS CONTROL THEORY?
1.1. Introduction
1.2. Systems
1.3. Schematics
1.4. Mathematical Systems
1.5. The Behavior of Systems
1.6. Improvement of the Behavior of Systems
1.7. More Detailed Breakdown
1.8. Uncertainty
1.9. Conclusion
BIBLIOGRAPHY AND COMMENTS
Chapter 2. SECOND-ORDER UNEAR DIFFERENTIAL AND DIFFERENCE EQUATIONS
2.1. Introduction
2.2. Second-Order Linear Differential Equations with Constant Coefficients
2.3. The Inhomogeneous Equation
2.4. Two-Point Boundary Conditions
2.5. First-Order Linear Differential Equations with Variable Coefficients
2.6. The Riccati Equation
2.7. Linear Equations with Variable Coefficients
2.8. The Inhomogeneous Equation
2.9. Green's Function
2.10. Linear Systems
2.11. Difference Equations
Miscellaneous Exercises
BIBLIOGRAPHY AND COMMENTS
Chapter 3. STABILITY AND CONTROL
3.1 Introduction
3.2. Stability
3.3. Numerical Solution and Stability
3.4. Perturbation Procedures
3.5. A Fundamental Stability Theorem
3.6. Stability by Design
3.7. Stability by Control
3.8. Proportional Control
3.9. Discussion
3.10. Analytic Formulation
3.11. One-Dimensional Systems
Miscellaneous Exercises
BIBLIOGRAPHY AND COMMENTS
Chapter 4. CONTINUOUS VARIATIONAL PROCESSES
CAICULUS OF VARIATIONS
4.1. Introduction
4.2. Does a Minimum Exist?
4.3. The Euler Equation
4.4. A Fallacious Argument
4.5. Haar's Device
4.6. Solution of the Euler Equation
4.7. Minimizing Property of the Solution
4.8. Alternative Approach
4.9. Asymptotic Control
4.10. Infinite Control Process
4.11. The Minimum Value of J(u)
4.12. Two-Point Constraints
4.13. Terminal Control
4.14. The Courant Parameter
4.15. Successive Approximations
4.16. min ?t0 [u2 + g(t)u2] dt
4.17. Discussion
4.18. The Simplicity of Control Processes
4.19. Discussion
4.20. The Minimum Value of j(u)
4.21. A Smoothing Process
4.22. Variation-Diminishing Property of Green's Function
4.23. Constraints
4.24. Minimizing Property
4.25. Monotonicity in ?
4.26. Proof of Monotonicity
4.27. Discussion
4.28. More General Quadratic Variational Problems
4.29. Variational Procedure
4.30. Proof of Minimum Property
4.31. Existence and Uniqueness
4.32. The Adjoint Operator
4.33. Sturm-Liouville Theory
4.34. Minimization by Means of Inequalities
4.35. Multiple Constraints
4.36. Unknown External Forces
Miscellaneous Exercises
BIBLIOGRAPHY AND COMMENTS
Chapter 5. DYNAMIC PROGRAMMING
5.1. Introduction
5.2. Control as a Multistage Decision Process
5.3. Preliminary Concepts
5.4. Formalism
5.5. Principle of Optimality
5.6. Discussion
5.7. Simplification
5.8. Validation
5.9. Infinite Process
5.10. Limiting Behavior as T8
5.11. Two-Point Boundary Problems
5.12. Time-Dependent Control Process
5.13. Global Constraints
5.14. Discrete Control Processes
5.15. Preliminaries
5.16. Recurrence Relation
5.17. Explicit Recurrence Relations
5.18. Behavior of rN
5.19. Approach to Steady-State Behavior
5.20. Equivalent Linear Relations
5.21. Local Constraints
5.22. Continuous as Limit of Discrete
5.23. Bang-Bang Control
5.24. Control in the Presence of Unknown Influences
5.25. Comparison between Calculus of Variations and Dynamic Programming
Miscellaneous Exercises
BIBLIOGRAPHY AND COMMENTS
Chapter 6. REVIEW OF MATRIX THEORY AND LINEAR DIFFERENTIAL EQUATIONS
6.1. Introduction
6.2. Vector-Matrix Notation
6.3. Inverse Matrix
6.4. The Product of Two Matrices
6.5. Inner Product and Norms
6.6. Orthogonal Matrices
6.7. Canonical Representation
6.8. Det A
6.9. Functions of a Symmetric Matrix
6.10. Positive Definite Matrices
6.11. Representation of A
6.12. Differentiation and Integration of Vectors and Matrices
6.13. The Matrix Exponential
6.14. Existence and Uniqueness Proof
6.15. Euler Technique and Asymptotic Behavior
6.16. x" - A(t)x = 0
6.17. x' = Ax + By, y' = Cx + Dy
6.18. Matrix Riccati Equation
6.19. dX/dt = AX + XB, X(0) = C
Miscellaneous Exercises
BIBLIOGRAPHY AND COMMENTS
Chapter 7. MULTIDIMENSIONAL CONTROL PROCESSES VIA THE CALCULUS OF VARIATIONS
7.1. Introduction
7.2. The Euler Equation
7.3. The Case of Constant A
7.4. Nonsingularity of cosh ST
7.5. The Minimum Value
7.6. Asymptotic Behavior
7.7. Variable A(t)
7.8. The Nonsingularity of X2'(T)
7.9. The Minimum Value
7.10. Computational Aspects
7.11. min ?T0 [(x, x) + (y, y)] dt, x' = Bx + y, x(0) = c
Miscellaneous Exercises
Chapter 8. MULTIDIMENSIONAL CONTROL PROCESSES VIA DYNAMIC PROGRAMMING
8.1. Introduction
8.2. ?T0 [(x', x') + (x, Ax)] dt
8.3. The Associated Riccati Equation
8.4. Asymptotic Behavior
8.5. Rigorous Aspects
8.6. Time-Dependent Case
8.7. Computational Aspects
8.8. Successive Approximations
8.9. Approximation in Policy Space
8.10. Monotone Convergence
8.11. Partitioning
8.12. Power Series Expansions
8.13. Extrapolation
8.14. Minimization via Inequalities
8.15. Discrete Control Processes
8.16. Ill-Conditioned Linear Systems
8.17. Lagrange Multipliers
8.18. Reduction of Dimensionality
8.19. Successive Approximations
8.20. Distributed Parameters
8.21. Slightly Intertwined Systems
Miscellaneous Exercises
BIBLIOGRAPHY AND COMMENTS
Chapter 9. FUNCTIONAL ANALYSIS
9.1. Motivation
9.2. The Hilbert Space L2(0, T)
9.3. Inner Products
9.4. Linear Operators
9.5. Vector Hilbert Space
9.6. Quadratic Functionals
9.7. Existence and Uniqueness of a Minimizing Function
9.8. The Equation for the Minimizing Function
9.9. Application to Differential Equations
9.10. Numerical Aspects
9.11. A Simple Algebraic Example
9.12. The Equation x + ?Bx = c
9.13. The Integral Equation ƒ(t) + ??T0 K(t, t1)ƒ(t1) dt1 = g(t)
9.14. Lagrange Multipliers
9.15. The Operator Ra
9.16. Control Subject to Constraints
9.17. Properties of f(c) and ?(c)
9.18. Statement of Result
Miscellaneous Exercises
BIBLIOGRAPHY AND COMMENTS
AUTHOR INDEX
SUBJECT INDEX

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Introduction to the Mathematical Theory of Control Processes

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