Introduction to the Mathematical Theory of Control Processes: Linear Equations and Quadratic Criteria v. 1
Published on 1. January 1967
244 pages
978-0-08-095547-6 (ISBN)
Description
This work discusses the theory of control processes. The extremely rapid growth of the theory, associated intimately with the continuing trend toward automation, makes it imperative that the courses of this nature rest upon a broad basis. The work discusses the fundamentals of the calculus of variations, dynamic programming, discrete control processes, use of the digital computer, and functional analysis. Introductory courses in control theory are essential for training the modern graduate student in pure and applied mathematics, engineering, mathematical physics, economics, biology, operations research, and related fields. The work also describes the dual approaches of the calculus of variations and dynamic programming in the scalar case and illustrates ways to tackle the multidimensional optimization problems.
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Richard Bellman
Introduction to the Mathematical Theory of Control Processes: Linear Equations and Quadratic Criteria v. 1
Book
02/1968
Academic Press
€72.43
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Content
- Front Cover
- Introduction to the Mathematical Theory of Control Processes
- Copyright Page
- 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 Linear 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: Calculus 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 ?0r[u'2 +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-1
- 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 ?0T [(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. ?0T [(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 f(t) + ? ?0T K(t, t1)f(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
- SUBJECT INDEX
