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1 Theory of Maxima and Minima＆THEODORE N．EDELBAUM 1

1．1 Necessary Conditions for Maxima or Minima 2

1．2 Sufficient Conditions for Maxima or Minima 4

1．3 Subsidiary Conditions 7

1．4 Application to Integrals 12

1．5 Remarks on Practical Application 16

1．6 Optimization of Low Thrust Trajectories and Propulsion Systems for a 24-Hour Equatorial Satellite 19

References 31

2 Direct Methods＆FRANK D．FAULKNER 33

2．0 Introduction and Summary 33

2．1 A Routine for Determining Some Optimum Trajectories 35

2．2 Elementary Graphic Solution 52

2．3 Optimum Thrust Programming along a Given Curve 59

References 65

3 Extremization of Linear Integrals by Green＇s Theorem＆ANGELO MIELE 69

3．1 Introduction 69

3．2 Linear Problem 70

3．3 Linear Isoperimetric Problem 76

3．4 Linear Problems in Flight Mechanics 77

3．5 Optimum Burning Program for a Short-Range,Nonlifting Missile 79

3．6 Optimum Drag Modulation Program for the Re-Entry of a Variable-Geometry Ballistic Missile 88

References 98

4 The Calculus of Variations in Applied Aerodynamics and Flight Mechanics＆ANGELO MIELE 99

4．1 Introduction 100

4．2 The Problem of Bolza 100

4．3 Transformation of Variational Problems 109

4．4 The Calculus of Variations in Applied Aerodynamics 112

4．5 Bodies of Revolution Having Minimum Pressure Drag in Newtonian Flow 113

4．6 Wings Having Minimum Pressure Drag in Linearized Supersonic Flow 120

4．7 The Calculus of Variations in Flight Mechanics 126

4．8 Optimum Trajectories for Rocket Flight in a Vacuum 127

4．9 Optimum Trajectories for Rocket Flight in a Resisting Medium 143

4．10 Conclusions 163

References 163

5 Variational Problems with Bounded Control Variables＆G．LEITMANN 171

5．0 Introduction 172

5．1 Statement of the Problem 172

5．2 Mass Flow Rate Limited Systems 174

5．3 Propulsive Power Limited Systems 182

5．4 Thrust Acceleration Limited Systems 186

5．5 Conclusions 192

5．6 Example 193

Nomenclature 200

Appendix 201

References 203

6 Methods of Gradients＆HENRY J．KELLEY 205

6．0 Introduction 206

6．1 Gradient Technique in Ordinary Minimum Problems 206

6．2 Gradient Technique in Flight Path Optimization Problems 216

6．3 Solar Sailing Example 233

6．4 Low-Thrust Example 241

6．5 Remarks on the Relative Merits of Various Computational Techniques 246

6．6 A Successive Approximation Scheme Employing the Min Operation 248

Appendix A 251

References 252

7 Pontryagin Maximum Principle＆RICHARD E．KOPP 255

7．0 Introduction 255

7．1 An Introduction to the Pontryagin Maximum Principle 256

7．2 The Adjoint System and the Pontryagin Maximum Principle 262

7．3 The Calculus of Variations and the Pontryagin Maximum Principle 264

7．4 Dynamic Programming and the Pontryagin Maximum Principle 271

7．5 Examples 274

References 278

8 On the Determination of Optimal Trajectories Via Dynamic Programming＆RICHARD BELLMAN 281

8．1 Introduction 281

8．2 Dynamic Programming 282

8．3 One-Dimensional Problems 282

8．4 Constraints 283

8．5 Constraints—II 283

8．6 Discussion 284

8．7 Two-Dimensional Problems 285

8．8 One-Dimensional Case 286

8．9 Discussion 288

8．10 Two-Dimensional Case 289

8．11 Discussion 290

References 290

9 Computational Considerations for Some Deterministic and Adaptive Control Processes＆ROBERT KALABA 291

9．1 Introduction 291

9．2 Some Deterministic Control Processes 292

9．3 Adaptive Control Processes 305

References 308

10 General Imbedding Theory＆C．M．KASHMAR AND E．L．PETERSON 311

10．1 Introduction 311

10．2 Problem Formulation 312

10．3 Elimination of Boundary Valuedness 313

10．4 Reduction to Final Value Problem 314

10．5 The Classical Solution 316

10．6 The Dynamic Programming Solution 318

10．7 Summary 321

References 321

11 Impulsive Transfer between Elliptical Orbits＆DEREK F．LAWDEN 323

11．1 Introduction 323

11．2 Impulsive Change in Orbital Elements 324

11．3 Dependence of Impulse on Orbital Elements 328

11．4 Optimal n-Impulse Transfer between Two Terminal Orbits 331

11．5 Optimal Two-Impulse Transfer 333

11．6 Optimal Slewing of the Orbital Axis 334

11．7 Transfer between Orbits Whose Axes Are Aligned 341

11．8 Appendix 349

References 350

12 The Optimum Spacing of Corrective Thrusts in Interplanetary Navigation＆JOHN BREAKWELL 353

12．1 Discussion and Results 353

12．2 Development of the Optimum Spacing in Example 1 363

12．3 Development of the Optimum Spacing in Example 2 365

12．4 The Covariance of Dn and Z>n+1 in the Case of Frequent Observations Since Launch;One-Dimensional Model 369

12．5 Estimation of D from Frequent Position Measurements since the Last Correction;One-Dimensional Model 371

Nomenclature 373

General References 375

13 Propulsive Efficiency of Rockets＆G．LEITMANN 377

13．0 Introduction 377

13．1 Propulsive Efficiency—Point Function 378

13．2 Propulsive Efficiency—Interval Function 381

Nomenclature 387

References 387

14 Some Topics in Nuclear Rocket Optimization＆R．W．BUSSARD 389

14．1 Introduction and Definition 389

14．2 High Acceleration Flight 396

14．3 Low Acceleration Flight 408

14．4 Heat Exchanger Propulsion Systems 428

14．5 Nuclear/Electric Propulsion Systems 440

References 445

AUTHOR INDEX 449

SUBJECT INDEX 452