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Introduction to the Mathematical Theory of Control Processes: Nonlinear Processes v. 2
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Introduction to the Mathematical Theory of Control Processes: Nonlinear Processes v. 2
Nonlinear Processes v. 2
Autorzy: Richard Bellman Wydawnictwo: Elsevier Science Data wydania: 1971 Język publikacji: Angielski Liczba stron: 329 Formaty publikacji: EAN: 9780080955483 ISBN: 9780080955483 Kategoria: Mathematical foundations Kolekcja specjalna: Rok matematyki 2019 Indeks wydawcy: S0076-5392(08)X6168-5 Nota bibliograficzna: -
Spis treści
- Front Cover 2
- Introduction to the Mathematical Theory of Control Processes 5
- Copyright Page 6
- Contents 9
- Preface 19
-
Chapter 1. Basic Concepts of Control Theory
23
- 1.1. Introduction 23
- 1.2. Systems and State Variables 24
- 1.3. Discussion 24
- 1.4. Control Variables 25
- 1.5. Criterion Function 26
- 1.6. Control Processes 27
- 1.7. Analytic Aspects 28
- 1.9. Event versus Time Orientation 29
- 1.8. Computational Aspects 29
- Front Cover 2
- Introduction to the Mathematical Theory of Control Processes 5
- Copyright Page 6
- Contents 9
- Preface 19
-
Chapter 1. Basic Concepts of Control Theory
23
- 1.1. Introduction 23
- 1.2. Systems and State Variables 24
- 1.3. Discussion 24
- 1.4. Control Variables 25
- 1.5. Criterion Function 26
- 1.6. Control Processes 27
- 1.7. Analytic Aspects 28
- 1.9. Event versus Time Orientation 29
- 1.8. Computational Aspects 29
- 1.10. Discrete versus Continuous 30
- Bibliography and Comments 30
-
Chapter 2. Discrete Control Processes and Dynamic Programming
32
- 2.1. Introduction 32
- 2.2. Existence of a Minimum 33
- 2.3. Uniqueness 34
- 2.4. Dynamic Programming Approach 34
- 2.5. Recurrence Relations 35
- 2.6. Imbedding 36
- 2.7. Policies 37
- 2.8. Principle of Optimality 38
- 2.9. Discussion 38
- 2.11. Constraints 39
- 2.10. Time Dependence 39
- 2.12. Analytic Aspects 40
- 2.13. Marginal Returns 41
- 2.14. Linear Equations and Quadratic Criteria 42
- Exercises 43
- 2.15. Discussion 44
- Exercise 44
- 2.16. An Example of Constraints 45
- 2.17. Summary of Results 48
- Exercises 48
- 2.18. Lagrange Parameter 49
- 2.19. Discussion 50
- 2.20. Courant Parameter 51
- Exercises 51
- Exercises 51
- 2.21. Infinite Processes 51
- 2.22. Approximation in Policy Space 52
- Exercises 53
- 2.24. Discrete Stochastic Control Processes 53
- 2.23. Minimum of Maximum Deviation 53
- Miscellaneous Exercises 55
- Bibliography and Comments 59
-
Chapter 3. Computational Aspects of Dynamic Programming
62
- 3.1. Introduction 62
- 3.2. Discretization 63
- 3.3. Minimization 64
- 3.4. Storage Requirements : Discretion in the Use of Discretization 65
- 3.5. Time Requirements 67
- 3.7. Simplified Policies 68
- 3.6. Discussion 68
- 3.9. Computing as a Control Process 71
- 3.8. Stability 71
- 3.10. Stability Analysis 72
- 3.11. Functions and Solutions 73
- Exercise 75
- 3.12. Interpolation 75
- 3.13. Computational Procedure 76
- 3.14. Evaluation of Polynomials 77
- 3.15. Orthogonal Polynomials and Quadrature 78
- 3.16. Polygonal Approximation 79
- 3.17. Adaptive Polygonal Approximation 80
- 3.18. Dynamic Programming Approach 81
- Exercises 82
- 3.20. Search Processes 83
- 3.19. Linkage 83
- Exercise 83
- 3.22. Continuity 85
- 3.21. Discussion 85
- Exercise 86
- 3.23. Restriction to Grid Points 86
- 3.24. Multidimensional Case 87
- Miscellaneous Exercises 87
- Bibliography and Comments 88
-
Chapter 4. Continuous Control Processes and the Calculus of Variations
92
- 4.1. Introduction 92
- 4.2. The Quadratic Case 94
- 4.3. Discussion 95
- 4.4. Formal Derivation of the Euler Equation 96
- 4.5. Haar's Device 97
- Exercises 98
- 4.6. Discussion 98
- 4.8. Nonexistence of a Minimum—II 99
- 4.7. Nonexistence of a Minimum—I 99
- Exercise 99
- Exercise 100
- 4.9. Nonexistence of a Minimum—III 100
- 4.10. Existence of Solution of Euler Equation 101
- 4.11. Successive Approximations 101
- Exercises 103
- 4.12. Conditional Uniqueness 104
- 4.13. A Plethora of Geodesics 104
- Exercises 105
- 4.14. A Priori Bounds 105
- 4.15. Convexity 106
- Exercises 106
- Exercises 107
- 4.16. Sufficient Condition for Absolute Minimum 107
- 4.17. Uniqueness of Solution of Euler Equation 108
- 4.18. Demonstration of Minimizing Property—Small T 108
- 4.19. Discussion 109
- 4.20. Solution as Function of Initial State 109
- 4.21. Solution as Function of Duration of Process 111
- Exercises 111
- 4.22. The Return Function 112
- 4.23. A Nonlinear Partial Differential Equation 112
- 4.24. Discussion 113
- Exercises 113
- 4.25. More General Control Processes 114
- 4.27. Multidimensional Control Processes—I 116
- 4.26. Discussion 116
- Exercises 116
- Exercise 117
- 4.28. Auxiliary Results 117
- Exercises 118
- 4.29. Multidimensional Control Processes—II 118
- Exercise 120
- 4.30. FunctionaI Analysis 120
- 4.31. Existence and Uniqueness of Solution of Euler Equation 121
- 4.32. Global Constraints 122
- Exercise 123
- 4.33. Necessity of Euler Equation 123
- 4.34. Minimization by Inequalities 124
- Exercise 125
- 4.35. Inverse Problems 126
- Miscellaneous Exercises 126
- Bibliography and Comments 127
-
Chapter 5. Computational Aspects of the Calculus of Variations
130
- 5.1. Introduction 130
- 5.2. Successive Approximations and Storage 131
- 5.3. Circumvention of Storage 132
- 5.4. Discussion 134
- 5.5. Functions and Algorithms 135
- 5.6. Quasilinearization 136
- 5.7. Convergence 137
- 5.8. Discussion 139
- 5.9. Judicious Choice of Initial Approximation 139
- 5.11. Multidimensional Case 140
- 5.10. Circumvention of Storage 140
- 5.12. Two-point Boundary-value Problems 141
- 5.13. Analysis of Computational Procedure 142
- 5.14. Instability 142
- 5.15. Dimensionality 143
- 5.16. Matrix Inversion 144
- 5.17. Tychonov Regularization 144
- 5.18. Quadrature Techniques 146
- 5.19. "The Proof Is in the Program" 147
- 5.21. Interpolation and Search 148
- 5.20. Numerical Solution of Nonlinear Euler Equations 148
- 5.22. Extrapolation 149
- 5.23. Bubnov-Galerkin Method 150
- 5.24. Method of Moments 151
- 5.25. Gradient Methods 152
- 5.26. A Specific Example 153
- 5.27. Numerical Procedure : Semi-discretization 154
- 5.28. Nonlinear Extrapolation 155
- 5.29. Rayleigh-Ritz Methods 156
- Miscellaneous Exercises 156
- Bibliography and Comments 157
-
Chapter 6. Continuous Control Processes and Dynamic Programming
160
- 6.1. Introduction 160
- 6.2. Continuous Multistage Decision Processes 161
- 6.3. Duality 162
- 6.4. Analytic Formalism 162
- 6.5. Limiting Form 164
- 6.6. Discussion 164
- 6.7. Associated Nonlinear Partial Differential Equations 164
- 6.8. Characteristics and the Euler Equation 166
- 6.9. Rigorous Aspects 166
- 6.11. Riccati Equation 168
- 6.10. Multidimensional Case 168
- 6.13. Finite Difference Techniques 170
- 6.12. Computational Significance 170
- 6.14. Unconventional Difference Approximations 172
- 6.15 . Unconventional Difference Approximations Continued 172
- 6.16. Power Series Expansions 173
- 6.17. Perturbation Series 174
- 6.19. Positivity 175
- 6.18. Existence and Uniqueness 175
- 6.21. Approximation in Policy Space 176
- 6.20. Uniqueness 176
- 6.22. Quasilinearization 177
- 6.23. Representation of Solution of Partial Differential Equation 178
- 6.24. Constraints 180
- 6.25. Inverse Problems 180
- Miscellaneous Exercises 182
- 6.26. Semi-groups and the Calculus of Variations 182
- Bibliography and Comments 185
-
Chapter 7. Limiting Behavior of Discrete Processes
189
- 7.1. Introduction 189
- 7.3. Suboptimization 190
- 7.2. Discrete Approximation to the Continuous and Conversely 190
- 7.4. Lower Bound 191
- 7.6. Linear Equations and Quadratic Criteria 192
- 7.7. Sophisticated Quadrature 192
- 7.5. Further Reduction 192
- 7.8. Degree of Approximation 193
- 7.9. Quadratic Case 194
- 7.10. Convex Case 195
- 7.11. Deferred Passage to the Limit 196
- 7.12. Use of Analytic Structure 197
- 7.14. Precise Formulation 198
- 7.13. Self-consistent Convergence 198
- 7.15. Lipschitz Conditions 199
- 7.16. An Intermediate Process 200
- 7.19. Almost Monotone Convergence 201
- 7.17. Comparison-of gn and h2n 201
- 7.18. fN(c, Δ) as a Function of Δ 201
- 7.20. Discussion 202
- Bibliography and Comments 202
-
Chapter 8. Asymptotic Control Theory
203
- 8.1. Introduction 203
- 8.2. Asymptotic Control 204
- 8.3. Existence of Limit of f(c,T) 205
- 8.4. Poincare-Lyapunov Theory 206
- 8.5. Analogous Result for Two-point Boundary-value Problem 207
- 8.6. The Associated Green's Function 208
- 8.7. Conversion to Integral Equation 210
- 8.8. Conditional Uniqueness 211
- 8.10. Asymptotic Behavior of u—II 212
- 8.9. Asymptotic Behavior of u—I 212
- 8.11. Infinite Control Process 214
- 8.12. Multidimensional Case 214
- 8.13. Asymptotic Control 215
- 8.14. Boundedness of ||x(t)|| 215
- 8.15. Convergence of f(c,T) 216
- 8.16. Conclusion of Proof 217
- 8.17. Infinite Processes 217
- 8.18. Discrete Infinite Processes 218
- 8.19. Steady-state Average Behavior 219
- 8.20. Subadditive Functions 220
- 8.21. Proof of Theorem 221
- Bibliography and Comments 223
-
Chapter 9. Duality and Upper and Lower Bounds
224
- 9.1. Introduction 224
- 9.2. Formalism 225
- 9.3. Quadratic Case 226
- 9.4. Multidimensional Quadratic Case 228
- 9.5. Numerical Utilization 229
- 9.6. The Legendre-Fenchel Transform 230
- 9.7. Convex g 231
- 9.8. Multidimensional Legendre-Fenchel Transform 232
- 9.9. Convex g(x) 233
- 9.10. Alternate Approach 234
- 9.11. Duality 235
- 9.12. Upper and Lower Bounds for Partial Differential Equations 236
- 9.13. Upper Bounds for f(c,T) 236
- 9.14. Lower Bounds 237
- 9.15. Perturbation Technique 238
- 9.17. 2g(c) = 2, h Convex 239
- 9.16. Convexity of h 239
- 9.18. The Maximum Transform 240
- 9.19. Application to Allocation Processes 241
- 9.20. Multistage Allocation 242
- 9.21. General Maximum Convolution 243
- Miscellaneous Exercises 244
- Bibliography and Comments 246
-
Chapter 10. Abstract Control Processes and Routing
248
- 10.1. Introduction 248
- 10.2. The Routing Problem 249
- 10.3. Dynamic Programming Approach 250
- 10.4. Upper and Lower Bounds 251
- 10.5. Existence and Uniqueness 253
- 10.6. Optimal Policy 255
- 10.7. Approximation in Policy Space 255
- 10.8. Computational Feasibility 256
- 10.9. Storage of Algorithms 257
- 10.10. Alternate Approaches 257
- 10.11. "Traveling Salesman'' Problem 258
- 10.12. Stratification 259
- 10.13. Routing and Control Processes 260
- 10.14. Computational Procedure 261
- 10.15. Feasibility 262
- 10.16. Perturbation Technique 263
- 10.17. Generalized Routing 264
- 10.18. Pawn-King Endings in Chess 265
- 10.19. Discussion 266
- Bibliography and Comments 267
-
Chapter 11. Reduction of Dimensionality
270
- 11.1. Introduction 270
- 11.2. A Terminal Control Process 271
- 11.4. New State Variables 272
- 11.3. Preliminary Transformation 272
- 11.5. Partial Differential Equation for ø(z,T) 273
- 11.6. Discussion 274
- 11.7. Riccati Differential Equation 274
- 11.9. Constraints 275
- 11.8. General Terminal Criterion 275
- 11.10. Successive Approximations 276
- 11.11. Quadratic Case 277
- 11.12. A General Nonlinear Case 278
- Bibliography and Comments 279
-
Chapter 12. Distributed Control Processes and the Calculus of Variations
281
- 12.1. Introduction 281
- 12.3. The Euler Formalism 282
- 12.2. A Heat Control Process 282
- 12.4. Rigorous Aspects 284
- 12.5. Laplace Transform 285