The Calculus of Variations and Optimal Control

1. Front Matter

   Pages i-xvi

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2. Calculus of Variations

   1. Front Matter

      Pages 1-1

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   2. Introduction

      • George Leitmann

      Pages 3-5

   3. Problem Statement and Necessary Conditions for an Extremum

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      Pages 7-24

   4. Integration of the Euler—Lagrange Equation

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      Pages 25-37

   5. An Inverse Problem

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      Pages 39-45

   6. The Weierstrass Necessary Condition

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      Pages 47-53

   7. Jacobi’s Necessary Condition

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      Pages 55-65

   8. Corner Conditions

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      Pages 67-70

   9. Concluding Remarks

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      Pages 71-73

3. Optimal Control

   1. Front Matter

      Pages 75-75

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   2. Introduction

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      Pages 77-78

   3. Problem Statement and Optimality

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      Pages 79-98

   4. Regular Optimal Trajectories

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      Pages 99-123

   5. Examples of Extremal Control

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      Pages 125-138

   6. Some Generalizations

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      Pages 139-209

   7. Special Systems

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      Pages 211-239

   8. Sufficient Conditions

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      Pages 241-263

   9. Feedback Control

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      Pages 265-284

   10. Optimization with Vector — Valued Cost

       • George Leitmann

       Pages 285-299

When the Tyrian princess Dido landed on the North African shore of the Mediterranean sea she was welcomed by a local chieftain. He offered her all the land that she could enclose between the shoreline and a rope of knotted cowhide. While the legend does not tell us, we may assume that Princess Dido arrived at the correct solution by stretching the rope into the shape of a circular arc and thereby maximized the area of the land upon which she was to found Carthage. This story of the founding of Carthage is apocryphal. Nonetheless it is probably the first account of a problem of the kind that inspired an entire mathematical discipline, the calculus of variations and its extensions such as the theory of optimal control. This book is intended to present an introductory treatment of the calculus of variations in Part I and of optimal control theory in Part II. The discussion in Part I is restricted to the simplest problem of the calculus of variations. The topic is entirely classical; all of the basic theory had been developed before the turn of the century. Consequently the material comes from many sources; however, those most useful to me have been the books of Oskar Bolza and of George M. Ewing. Part II is devoted to the elementary aspects of the modern extension of the calculus of variations, the theory of optimal control of dynamical systems.

• Calculus of Variations
• Optimal control
• calculus
• control
• differential equation
• extrema
• optimization

• University of California, Berkeley, USA

  George Leitmann

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