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CHAPTER Ⅰ FEEDBACK CONTROL AND THE CALCULUS OF VARIATIONS13

1.1 Introduction13

1.2 Mathematical description of a physical system13

1.3 Parenthetical15

1.4 Hereditary influences16

1.5 Criteria of performance17

1.6 Terminal control18

1.7 Control process18

1.8 Feedback control19

1.9 An alternate concept20

1.10 Feedback control as a variational problem21

1.11 The scalar variational problem22

1.12 Discussion24

1.13 Relative minimum versus absolute minimum24

1.14 Nonlinear differential equations26

1.15 Two-point boundary value problems27

1.16 An example of multiplicity of solution29

1.17 Non-analytic criteria30

1.18 Terminal control and implicit variational problems31

1.19 Constraints32

1.20 Linearity34

1.21 Summing up35

Bibliography and discussion36

CHAPTER Ⅱ DYNAMICAL SYSTEMS AND TRANSFORMATIONS41

2.1 Introduction41

2.2 Functions of initial values41

2.3 The principle of causality42

2.4 The basic functional equation42

2.5 Continuous version43

2.6 The functional equations satisfied by the elementary functions43

2.7 The matrix exponential44

2.8 Transformations and iteration44

2.9 Carleman's linearization45

2.10 Functional equations and maximum range46

2.11 Vertical motion—Ⅰ46

2.12 Vertical motion—Ⅱ47

2.13 Maximum altitude47

2.14 Maximum range48

2.15 Multistage processes and differential equations48

Bibliography and discussion48

CHAPTER Ⅲ MULTISTAGE DECISION PROCESSES AND DYNAMIC PROGRAMMING51

3.1 Introduction51

3.2 Multistage decision processes51

3.3 Discrete deterministic multistage decision processes52

3.4 Formulation as a conventional maximization problem53

3.5 Markovian-type processes54

3.6 Dynamic programming approach55

3.7 A recurrence relation56

3.8 The principle of optimality56

3.9 Derivation of recurrence relation57

3.10 “Terminal” control57

3.11 Continuous deterministic processes58

3.12 Discussion59

Bibliography and comments59

CHAPTER Ⅳ DYNAMIC PROGRAMMING AND THE CALCULUS OF VARIATIONS61

4.1 Introduction61

4.2 The calculus of variations as a multistage decision process62

4.3 Geometric interpretation62

4.4 Functional equations63

4.5 Limiting partial differential equations64

4.6 The Euler equations and characteristics65

4.7 Discussion66

4.8 Direct derivation of Euler equation66

4.9 Discussion67

4.10 Discrete processes67

4.11 Functional equations68

4.12 Minimum of maximum deviation69

4.13 Constraints70

4.14 Structure of optimal policy70

4.15 Bang-bang control72

4.16 Optimal trajectory73

4.17 The brachistochrone74

4.18 Numerical computation of solutions of differential equations76

4.19 Sequential computation77

4.20 An example78

Bibliography and comments79

CHAPTER Ⅴ COMPUTATIONAL ASPECTS OF DYNAMIC PROGRAMMING85

5.1 Introduction85

5.2 The computational process—Ⅰ86

5.3 The computational process—Ⅱ87

5.4 The computational process—Ⅲ88

5.5 Expanding grid88

5.6 The computational process—Ⅳ88

5.7 Obtaining the solution from the numerical results89

5.8 Why is dynamic programming better than straightforward enumeration90

5.9 Advantages of dynamic programming approach90

5.10 Absolute maximum versus relative maximum91

5.11 Initial value versus two-point boundary value problems91

5.12 Constraints91

5.13 Non-analyticity92

5.14 Implicit variational problems92

5.15 Approximation in policy space93

5.16 The curse of dimensionality94

5.17 Sequential search95

5.18 Sensitivity analysis95

5.19 Numerical solution of partial differential equations95

5.20 A simple nonlinear hyperbolic equation96

5.21 The equation fT=g1+g2fc+g3fc297

Bibliography and comments98

CHAPTER Ⅵ THE LAGRANGE MULTIPLIER100

6.1 Introduction100

6.2 Integral constraints101

6.3 Lagrange multiplier102

6.4 Discussion103

6.5 Several constraints103

6.6 Discussion104

6.7 Motivation for the Lagrange multiplier105

6.8 Geometric motivation106

6.9 Equivalence of solution108

6.10 Discussion109

Bibliography and discussion110

CHAPTER Ⅶ TWO-POINT BOUNDARY VALUE PROBLEMS111

7.1 Introduction111

7.2 Two-point boundary value problems112

7.3 Application of dynamic programming techniques113

7.4 Fixed terminal state113

7.5 Fixed terminal state and constraint115

7.6 Fixed terminal set115

7.7 Internal conditions116

7.8 Characteristic value problems116

Bibliography and comments117

CHAPTER Ⅷ SEQUENTIAL MACHINES AND THE SYNTHESIS OF LOGICAL SYSTEMS119

8.1 Introduction119

8.2 Sequential machines119

8.3 Information pattern120

8.4 Ambiguity121

8.5 Functional equations121

8.6 Limiting case122

8.7 Discussion122

8.8 Minimum time123

8.9 The coin-weighing problem123

8.10 Synthesis of logical systems124

8.11 Description of problem124

8.12 Discussion125

8.13 Introduction of a norm125

8.14 Dynamic programming approach125

8.15 Minimum number of stages125

8.16 Medical diagnosis126

Bibliography and discussion126

CHAPTER Ⅸ UNCERTAINTY AND RANDOM PROCESSES129

9.1 Introduction129

9.2 Uncertainty130

9.3 Sour grapes or truth?131

9.4 Probability132

9.5 Enumeration of equally likely possibilities132

9.6 The frequency approach134

9.7 Ergodic theory136

9.8 Random variables136

9.9 Continuous stochastic variable137

9.10 Generation of random variables138

9.11 Stochastic process138

9.12 Linear stochastic sequences138

9.13 Causality and the Markovian property139

9.14 Chapman-Kolmogoroff equations140

9.15 The forward equations141

9.16 Diffusion equations142

9.17 Expected values142

9.18 Functional equations144

9.19 An application145

9.20 Expected range and altitude145

Bibliography and comments146

CHAPTER Ⅹ STOCHASTIC CONTROL PROCESSES152

10.1 Introduction152

10.2 Discrete stochastic multistage decision processes152

10.3 The optimization problem153

10.4 What constitutes an optimal policy?154

10.5 Two particular stochastic control processes155

10.6 Functional equations155

10.7 Discussion156

10.8 Terminal control156

10.9 Implicit criteria157

10.10 A two-dimensional process with implicit criterion157

Bibliography and discussion158

CHAPTER Ⅺ MARKOVIAN DECISION PROCESSES160

11.1 Introduction160

11.2 Limiting behavior of Markov processes160

11.3 Markovian decision processes—Ⅰ161

11.4 Markovian decision processes—Ⅱ162

11.5 Steady-state behavior162

11.6 The steady-state equation163

11.7 Howard's iteration scheme164

11.8 Linear programming and sequential decision processes164

Bibliography and comments165

CHAPTER Ⅻ QUASILINEARIZATION167

12.1 Introduction167

12.2 Continuous Markovian decision processes168

12.3 Approximation in policy space168

12.4 Systems170

12.5 The Riccati equation170

12.6 Extensions171

12.7 Monotone approximation in the calculus of variations171

12.8 Computational aspects172

12.9 Two-point boundary-value problems173

12.10 Partial differential equations174

Bibliography and comments175

CHAPTER ⅩⅢ STOCHASTIC LEARNING MODELS176

13.1 Introduction176

13.2 A stochastic learning model176

13.3 Functional equations177

13.4 Analytic and computational aspects177

13.5 A stochastic learning model—Ⅱ177

13.6 Inverse problem178

Bibliography and comments178

CHAPTER ⅩⅣ THE THEORY OF GAMES AND PURSUIT PROCESSES180

14.1 Introduction180

14.2 A two-person process181

14.3 Multistage process181

14.4 Discussion182

14.5 Borel-von Neumann theory of games182

14.6 The min-max theorem of von Neumann184

14.7 Discussion184

14.8 Computational aspects185

14.9 Card games185

14.10 Games of survival185

14.11 Control processes as games against nature186

14.12 Pursuit processes—minimum time to capture187

14.13 Pursuit processes—minimum miss distance189

14.14 Pursuit processes—minimum miss distance within a given time189

14.15 Discussion190

Bibliography and discussion190

CHAPTER ⅩⅤ ADAPTIVE PROCESSES194

15.1 Introduction194

15.2 Uncertainty revisited195

15.3 Reprise198

15.4 Unknown—Ⅰ199

15.5 Unknown—Ⅱ200

15.6 Unknown—Ⅲ200

15.7 Adaptive processes201

Bibliography and comments201

CHAPTER ⅩⅥ ADAPTIVE CONTROL PROCESSES203

16.1 Introduction203

16.2 Information pattern205

16.3 Basic assumptions207

16.4 Mathematical formulation208

16.5 Functional equations208

16.6 From information patterns to distribution functions209

16.7 Feedback control209

16.8 Functional equations210

16.9 Further structural assumptions210

16.10 Reduction from functionals to functions211

16.11 An illustrative example—deterministic version211

16.12 Stochastic version212

16.13 Adaptive version213

16.14 Sufficient statistics215

16.15 The two-armed bandit problem215

Bibliography and discussion216

CHAPTER ⅩⅦ SOME ASPECTS OF COMMUNICATION THEORY219

17.1 Introduction219

17.2 A model of a communication process220

17.3 Utility a function of use221

17.4 A stochastic allocation process221

17.5 More general processes222

17.6 The efficient gambler222

17.7 Dynamic programming approach223

17.8 Utility of a communication channel224

17.9 Time-dependent case224

17.10 Correlation225

17.11 M-signal channels225

17.12 Continuum of signals227

17.13 Random duration228

17.14 Adaptive processes228

Bibliography and comments230

CHAPTER ⅩⅧ SUCCESSIVE APPROXIMATION232

18.1 Introduction232

18.2 The classical method of successive approximations233

18.3 Application to dynamic programming234

18.4 Approximation in policy space235

18.5 Quasilinearization236

18.6 Application of the preceding ideas236

18.7 Successive approximations in the calculus of variations237

18.8 Preliminaries on differential equations239

18.9 A terminal control process240

18.10 Differential-difference equations and retarded control241

18.11 Quadratic criteria241

18.12 Successive approximations once more243

18.13 Successive approximations—Ⅱ244

18.14 Functional approximation244

18.15 Simulation techniques246

18.16 Quasi-optimal policies246

18.17 Non-zero sum games247

18.18 Discussion248

Bibliography and discussion249