Introduction to the Mathematical Theory of Control Processes: Nonlinear Processes v. 2

Name: Introduction to the Mathematical Theory of Control Processes: Nonlinear Processes v. 2
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Academic Press

Published on 20. April 1971

305 pages

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Front Cover
Introduction to the Mathematical Theory of Control Processes
Copyright Page
Contents
Preface
Chapter 1. Basic Concepts of Control Theory
1.1. Introduction
1.2. Systems and State Variables
1.3. Discussion
1.4. Control Variables
1.5. Criterion Function
1.6. Control Processes
1.7. Analytic Aspects
1.8. Computational Aspects
1.9. Event versus Time Orientation
1.10. Discrete versus Continuous
Bibliography and Comments
Chapter 2. Discrete Control Processes and Dynamic Programming
2.1. Introduction
2.2. Existence of a Minimum
2.3. Uniqueness
2.4. Dynamic Programming Approach
2.5. Recurrence Relations
2.6. Imbedding
2.7. Policies
2.8. Principle of Optimality
2.9. Discussion
2.10. Time Dependence
2.11. Constraints
2.12. Analytic Aspects
2.13. Marginal Returns
2.14. Linear Equations and Quadratic Criteria
Exercises
2.15. Discussion
Exercise
2.16. An Example of Constraints
2.17. Summary of Results
Exercises
2.18. Lagrange Parameter
2.19. Discussion
Exercises
2.20. Courant Parameter
Exercises
2.21. Infinite Processes
2.22. Approximation in Policy Space
2.23. Minimum of Maximum Deviation
Exercises
2.24. Discrete Stochastic Control Processes
Miscellaneous Exercises
Bibliography and Comments
Chapter 3. Computational Aspects of Dynamic Programming
3.1. Introduction
3.2. Discretization
3.3. Minimization
3.4. Storage Requirements : Discretion in the Use of Discretization
3.5. Time Requirements
3.6. Discussion
3.7. Simplified Policies
3.8. Stability
3.9. Computing as a Control Process
3.10. Stability Analysis
3.11. Functions and Solutions
Exercise
3.12. Interpolation
3.13. Computational Procedure
3.14. Evaluation of Polynomials
3.15. Orthogonal Polynomials and Quadrature
3.16. Polygonal Approximation
3.17. Adaptive Polygonal Approximation
3.18. Dynamic Programming Approach
Exercises
3.19. Linkage
Exercise
3.20. Search Processes
3.21. Discussion
3.22. Continuity
Exercise
3.23. Restriction to Grid Points
3.24. Multidimensional Case
Miscellaneous Exercises
Bibliography and Comments
Chapter 4. Continuous Control Processes and the Calculus of Variations
4.1. Introduction
4.2. The Quadratic Case
4.3. Discussion
4.4. Formal Derivation of the Euler Equation
4.5. Haar's Device
Exercises
4.6. Discussion
4.7. Nonexistence of a Minimum-I
Exercise
4.8. Nonexistence of a Minimum-II
Exercise
4.9. Nonexistence of a Minimum-III
4.10. Existence of Solution of Euler Equation
4.11. Successive Approximations
Exercises
4.12. Conditional Uniqueness
4.13. A Plethora of Geodesics
Exercises
4.14. A Priori Bounds
Exercises
4.15. Convexity
Exercises
4.16. Sufficient Condition for Absolute Minimum
4.17. Uniqueness of Solution of Euler Equation
4.18. Demonstration of Minimizing Property-Small T
4.19. Discussion
4.20. Solution as Function of Initial State
Exercises
4.21. Solution as Function of Duration of Process
4.22. The Return Function
4.23. A Nonlinear Partial Differential Equation
4.24. Discussion
Exercises
4.25. More General Control Processes
4.26. Discussion
Exercises
4.27. Multidimensional Control Processes-I
Exercise
4.28. Auxiliary Results
Exercises
4.29. Multidimensional Control Processes-II
Exercise
4.30. FunctionaI Analysis
4.31. Existence and Uniqueness of Solution of Euler Equation
4.32. Global Constraints
Exercise
4.33. Necessity of Euler Equation
4.34. Minimization by Inequalities
Exercise
4.35. Inverse Problems
Miscellaneous Exercises
Bibliography and Comments
Chapter 5. Computational Aspects of the Calculus of Variations
5.1. Introduction
5.2. Successive Approximations and Storage
5.3. Circumvention of Storage
5.4. Discussion
5.5. Functions and Algorithms
5.6. Quasilinearization
5.7. Convergence
5.8. Discussion
5.9. Judicious Choice of Initial Approximation
5.10. Circumvention of Storage
5.11. Multidimensional Case
5.12. Two-point Boundary-value Problems
5.13. Analysis of Computational Procedure
5.14. Instability
5.15. Dimensionality
5.16. Matrix Inversion
5.17. Tychonov Regularization
5.18. Quadrature Techniques
5.19. "The Proof Is in the Program
5.20. Numerical Solution of Nonlinear Euler Equations
5.21. Interpolation and Search
5.22. Extrapolation
5.23. Bubnov-Galerkin Method
5.24. Method of Moments
5.25. Gradient Methods
5.26. A Specific Example
5.27. Numerical Procedure : Semi-discretization
5.28. Nonlinear Extrapolation
5.29. Rayleigh-Ritz Methods
Miscellaneous Exercises
Bibliography and Comments
Chapter 6. Continuous Control Processes and Dynamic Programming
6.1. Introduction
6.2. Continuous Multistage Decision Processes
6.3. Duality
6.4. Analytic Formalism
6.5. Limiting Form
6.6. Discussion
6.7. Associated Nonlinear Partial Differential Equations
6.8. Characteristics and the Euler Equation
6.9. Rigorous Aspects
6.10. Multidimensional Case
6.11. Riccati Equation
6.12. Computational Significance
6.13. Finite Difference Techniques
6.14. Unconventional Difference Approximations
6.15 . Unconventional Difference Approximations Continued
6.16. Power Series Expansions
6.17. Perturbation Series
6.18. Existence and Uniqueness
6.19. Positivity
6.20. Uniqueness
6.21. Approximation in Policy Space
6.22. Quasilinearization
6.23. Representation of Solution of Partial Differential Equation
6.24. Constraints
6.25. Inverse Problems
6.26. Semi-groups and the Calculus of Variations
Miscellaneous Exercises
Bibliography and Comments
Chapter 7. Limiting Behavior of Discrete Processes
7.1. Introduction
7.2. Discrete Approximation to the Continuous and Conversely
7.3. Suboptimization
7.4. Lower Bound
7.5. Further Reduction
7.6. Linear Equations and Quadratic Criteria
7.7. Sophisticated Quadrature
7.8. Degree of Approximation
7.9. Quadratic Case
7.10. Convex Case
7.11. Deferred Passage to the Limit
7.12. Use of Analytic Structure
7.13. Self-consistent Convergence
7.14. Precise Formulation
7.15. Lipschitz Conditions
7.16. An Intermediate Process
7.17. Comparison-of gn and h2n
7.18. fN(c, ?) as a Function of ?
7.19. Almost Monotone Convergence
7.20. Discussion
Bibliography and Comments
Chapter 8. Asymptotic Control Theory
8.1. Introduction
8.2. Asymptotic Control
8.3. Existence of Limit of f(c,T)
8.4. Poincare-Lyapunov Theory
8.5. Analogous Result for Two-point Boundary-value Problem
8.6. The Associated Green's Function
8.7. Conversion to Integral Equation
8.8. Conditional Uniqueness
8.9. Asymptotic Behavior of u-I
8.10. Asymptotic Behavior of u-II
8.11. Infinite Control Process
8.12. Multidimensional Case
8.13. Asymptotic Control
8.14. Boundedness of ||x(t)||
8.15. Convergence of f(c,T)
8.16. Conclusion of Proof
8.17. Infinite Processes
8.18. Discrete Infinite Processes
8.19. Steady-state Average Behavior
8.20. Subadditive Functions
8.21. Proof of Theorem
Bibliography and Comments
Chapter 9. Duality and Upper and Lower Bounds
9.1. Introduction
9.2. Formalism
9.3. Quadratic Case
9.4. Multidimensional Quadratic Case
9.5. Numerical Utilization
9.6. The Legendre-Fenchel Transform
9.7. Convex g
9.8. Multidimensional Legendre-Fenchel Transform
9.9. Convex g(x)
9.10. Alternate Approach
9.11. Duality
9.12. Upper and Lower Bounds for Partial Differential Equations
9.13. Upper Bounds for f(c,T)
9.14. Lower Bounds
9.15. Perturbation Technique
9.16. Convexity of h
9.17. 2g(c) = 2, h Convex
9.18. The Maximum Transform
9.19. Application to Allocation Processes
9.20. Multistage Allocation
9.21. General Maximum Convolution
Miscellaneous Exercises
Bibliography and Comments
Chapter 10. Abstract Control Processes and Routing
10.1. Introduction
10.2. The Routing Problem
10.3. Dynamic Programming Approach
10.4. Upper and Lower Bounds
10.5. Existence and Uniqueness
10.6. Optimal Policy
10.7. Approximation in Policy Space
10.8. Computational Feasibility
10.9. Storage of Algorithms
10.10. Alternate Approaches
10.11. "Traveling Salesman'' Problem
10.12. Stratification
10.13. Routing and Control Processes
10.14. Computational Procedure
10.15. Feasibility
10.16. Perturbation Technique
10.17. Generalized Routing
10.18. Pawn-King Endings in Chess
10.19. Discussion
Bibliography and Comments
Chapter 11. Reduction of Dimensionality
11.1. Introduction
11.2. A Terminal Control Process
11.3. Preliminary Transformation
11.4. New State Variables
11.5. Partial Differential Equation for ø(z,T)
11.6. Discussion
11.7. Riccati Differential Equation
11.8. General Terminal Criterion
11.9. Constraints
11.10. Successive Approximations
11.11. Quadratic Case
11.12. A General Nonlinear Case
Bibliography and Comments
Chapter 12. Distributed Control Processes and the Calculus of Variations
12.1. Introduction
12.2. A Heat Control Process
12.3. The Euler Formalism
12.4. Rigorous Aspects
12.5. Laplace Transform
12.6. An Integral Equation
12.7. The Variational Equation
12.8. Computational Approaches
12.9. Modified Liouviile-Neumann Solution
12.10. Use of Difference Approximations
12.11. Generalized Quadrature
12.12. Orthogonal Expansion and Truncation
12,13. Differential-Difference Equation
12.14. The Euler Equation
12.15. Discussion
12.16. Laplace Transform
12.17. Integral Equation
12.18. Variational Equation
12.19. Discretization
12.20. Discussion
Miscellaneous Exercises
Bibliography and Comments
Chapter 13. Distributed Control Processes and Dynamic Programming
13.1. Introduction
13.2. Formulation of a General Control Process
13.3. Multistage Decision Process
13.4. Functional Derivative
13.5. Formal Derivation of Functional Equation
13.6. Discussion
13.7. Quadratic Criteria
13.8. Integral Equations
13.9. Expression for min J(u)
13.10. Functional Equation for f (v, a)
13.11. The Form of L(k(a, t), a)
13.12. Functional Equation for q(t, s, a)
13.13. The Potential Equation
13.14. Reduction of Dimensionality-I
13.15. Reduction in Dimensionality-I
Miscellaneous Exercises
Bibliography and Comments
Chapter 14. Some Directions of Research
14.1. Introduction
14.2. Constraints
14.3. Control Takes Time
14.4. Principle of Macroscopic Uncertainty
14.5. ''On-Line" Control
14.6. Monotone Approximation
14.7. Identification and Control
14.8. Mathematical Model-Making
14.9. Computing as a Control Process
14.10. Physiological Control Processes
14.11. Environmental Control
14.12. Classical and Nonclassical Mechanics
14.13. Two-Person and N-Person Processes
14.14. Conclusion
Bibliography and Comment
Author Index
Subject Index

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