Some Aspects of the mathematical theory of control processes:
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | Undetermined |
| Published: |
Santa Monica, Calif.
Rand Corp.
1958
|
| Series: | Project Rand
|
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIX, 244 Bl. |
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MARC
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| 100 | 1 | |a Bellman, Richard |d 1920-1984 |e Verfasser |0 (DE-588)120476568 |4 aut | |
| 245 | 1 | 0 | |a Some Aspects of the mathematical theory of control processes |c R. E. Bellman ; I. Glicksberg ; O. A. Gross |
| 264 | 1 | |a Santa Monica, Calif. |b Rand Corp. |c 1958 | |
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| 700 | 1 | |a Glicksberg, Irving |e Verfasser |4 aut | |
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Record in the Search Index
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| adam_text | COMTENTS
Summary iii
Preface ix
Part I
DIFFERENTIAL, DIFFERENCE, AND RELATED LINEAR
FUNCTIONAL EQUATIONS
Chapter
1. Preliminary Results 3
1.1. Introduction 3
1.2. Differential Equations—Notation 3
1.3- The Linear Equation with Constant Coefficients 4
1.4. The Inhomogeneous Linear Equation with Constant
Coefficients 6
1.5. The Inhomogeneous Equation with Variable Coefficients 7
1.6. Difference Equations 7
1.7. Differential-Difference Equations 8
1.8. Volterra Integral Equations 9
1.9. The Renewal Equation 10
1.10. Fredholm Integral Equations 10
1.11. Linear Operations 11
2. Stability Theory 13
2.1. Introduction :..... 13
2.2. The Linear Equation with Constant Coefficients 13
2.3. The Hurwitz Conditions for Stability 14
2.4. The Poincare-Lyapunov Stability Theorem 15
2.5. Difference Equations 16
2.6. Differential-Difference Equations 17
Part II
LINEAR AND QUADRATIC CONTROL PROBLEMS
3. Control Problems of the First Kind 21
3.1. Introduction 21
3.2. A Transformation of the Problem 23
xv
Chapter
3.3. The Neyman-Pearson Lemma 24
3.4. Another Approach 2ft
3.5. The Moment Space of Characteristic Functions 28
3.6. A Simple Example 31
3.7. Inequality Replaced by Equality 32
3.8. The Multidimensional Problem , 34
3.9- A Result Concerning Positiviry 35
3.10. A Maximization Problem Due to H. Markowitz 37
3.11. The One-dimensional Problem 37
3.12. The Two-dimensional Problem 38
3.13. The «-Dimensional Problem 41 .
4. Quadratic Functional 45
4.1. Introduction 45
4.2. Existence and Uniqueness of Solution 46
4.3. The Equation for the Minimizing Function 48
4.4. Properties of an Operator 49
4.5. Some Associated Problems 50
4.6. Statement of Results 52
4.7. A Useful Formula 53
4.8. Application to Differential Equations 53
4.9. The Solution of the Minimum Problem 55
4.10. Numerical Methods 56
4.11. A Simple Algebraic Example 56
4.12. The Solution of .-: -•- A/f.v = c 5?
4.13. The Solution of f(x) -*- A f1 K(x, y)t(y) dy - g(x) ... 5S
4.14. External Influences ..... 58
4.15. Quadratic Deviation 59
4.16. The Second-order Equation ¦. .. 60
4.17. The Hth-order Equation 61
4.18. Linear Systems 61
4.19. Hurwitz Conditions 62
Part III
VARIATIONAL PROBLEMS WITH CONSTRAINTS
5. Control Problems of the Third Kind 67
5.1. Introduction 67
5.2. Quadratic Deviation—Linear Cost 67
5.3. Linear Cost—Deviation Measured by Rate of Change. ... 71
5.4. The Function.1.! max . I - :¦:! 16
Chapter
6. Classical Variaiiunal Theory with Constraints 83
6.1. Introduction 83
6.2. Preliminary Reductions 84
6.3. Existence of a Solution 85
6.4. Necessary Conditions for a Maximum 88
6.5. Uniqueness 91
6.6. Continuity Properties of Solutions 92
6.7. Solutions of a Particular Type 94
6.8. The Unbounded Case 98
6.9. Examples 100
6.10. The w-Dirnensional Case 103
7. The Bang-Bang Control Problem 107
7.1. Introduction 107
. 7.2. Statement of Results 107
7.3. Proof of Theorem 107
7.4. A Special Case 110
Part IV
THE THEORY OF DYNAMIC PROGRAMMING
8. Preliminaries on the Theory of Dynamic Programming 119
8.1. Introduction 119
8.2. A Multistage Allocation Process 119
8.3. Discussion of Process 121
8.4. Some Terminology 121
8.5. Functional-equation Formulation 122
8.6. Sensitivity Analysis 122
8.7. The Principle of Opfimality 123
8.8. A General Formulation of Deterministic Dynamic-
programming Processes ¦.. ... 123
8.9. A Stochastic Multistage Decision Process 124
8.10. Approximation in Function Space 125
9- Dynamic Programming and the Calculus of Variations 129
9.1. Introduction 129
9.2. . The Basic Functional Equation 130
9.3. Heuristic Considerations 131
9.4. One Method of Proof 132
9.5. Finite Version 133
9.6. A Simple Case 134
9.7. A More General Problem 138
9.8. The Function fz{c) 139
Chapter
9.9. A Lemma on Concavity 140
9.10. The Function /3(c) 142
9-11. N-Dimensional Allocation Process 143
9.12. Time-dependent Process 144
9.13- Smoothing Processes 145
9.14. Computational Aspects—Allocation Process 146
9.15. Computational Aspects—Smoothing Process 147
9.16. Quadratic Criteria 148
9.17. Inventor) Cost 148
9.18. Control of a Physical System 149
9.19- Second-order System 150
9.20. Minimum of Maximum Deviation 151
9.21. Rocket-powered Aircraft and Optimal Climb 152
9.22. Maximum Range—Discrete Version 153
9-23. Minimum Time—Discrete Version 154
10. Optimal Inventor) and Stock Control 155
10.1. Introduction 155
10.2. Formulation of the General Problem 156
10.3. Existence and Uniqueness Theorems 159
10.4. A Simple Observation 162
10.5. Preliminaries 163
10.6. Proportional Cost—One-dimensional Case 163
10.7. Proportional Cost—Multidimensional Case 168
10.8. Finite Time Period 170
10.9. Finite Time—Multidimensional Case 173
10.10. Nonproportional Penalty Cost—Red Tape 174
10.11. Particular Cases : , 176
10.12. The Form of the General Solution 176
10.13. Preliminaries : ¦..:.• 177
10.14. Unbounded Process—One-period Time Lag 177
10.15. Convex Cost Function—Unbounded Process ISO
10.16. Successive Approximations 182
10.17. Obtaining a First Approximation 182
10.18. The Renewal Equation 182
10.19. The Laplace Transform 183
Part V
THE THEORY OF GAMES
11. Preliminaries on the Theory of Games 189
11.1. Introduction 189
Chapter
11.2. The Fundamental Min-Max Theorem 190
11.3. Continuous Games 190
11.4. Games over Function Spaces 191
11.5. Finite Resources and Multistage Games 191
11.6. Nonzero-sum Games 192
12. An Allocation Problem 195
12.1. Introduction 195
12.2. The Neyman-Pearson Lemma 196
12.3. The Solution for Strictly Monotone Functions 197
12.4. Extension 200
12.5. A Particular Case 201
13. Application of the Theory of Games to the Calculus of Variations. . 203
• 13.1. Introduction 203
13-2. The Basic Device 204
13.3. Min-Max Theorems 204
13.4. Consideration of min fvT | 1 — x(t) dt 205
13.5. Consideration of max j 1 — x(l) j 208
13-6. Discussion 209
14. The Diversions of Red Dog and Poker 211
14.1. Introduction 211
14.2. Red Dog 211
14.3. Blackjack 21,6
14.4. A Simple Poker Game 21?
14.5. A Poker Game with a Raise 222
15. Some Computational Algorithms 229
15.1. Introduction 229
15.2. The Iterative Procedure of Brown 229
15.3. The J. Robinson Proof of Convergence ..... 230
15.4. Continuous Iteration 234
15.5. The Limiting Case 237
15.6. The Corresponding Discrete Method 238
15.7: The Perron Root 238
15.8. An Alternative Definition 240
15.9. Nonlinear Recurrence Relation 242
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| spelling | Bellman, Richard 1920-1984 Verfasser (DE-588)120476568 aut Some Aspects of the mathematical theory of control processes R. E. Bellman ; I. Glicksberg ; O. A. Gross Santa Monica, Calif. Rand Corp. 1958 XIX, 244 Bl. txt rdacontent n rdamedia nc rdacarrier Project Rand Glicksberg, Irving Verfasser aut Gross, O. A. Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018499736&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bellman, Richard 1920-1984 Glicksberg, Irving Gross, O. A. Some Aspects of the mathematical theory of control processes |
| title | Some Aspects of the mathematical theory of control processes |
| title_auth | Some Aspects of the mathematical theory of control processes |
| title_exact_search | Some Aspects of the mathematical theory of control processes |
| title_full | Some Aspects of the mathematical theory of control processes R. E. Bellman ; I. Glicksberg ; O. A. Gross |
| title_fullStr | Some Aspects of the mathematical theory of control processes R. E. Bellman ; I. Glicksberg ; O. A. Gross |
| title_full_unstemmed | Some Aspects of the mathematical theory of control processes R. E. Bellman ; I. Glicksberg ; O. A. Gross |
| title_short | Some Aspects of the mathematical theory of control processes |
| title_sort | some aspects of the mathematical theory of control processes |
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