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The Calculus of Variations and Optimal Control

An Introduction

  • Book
  • © 1981

Overview

Authors:
  1. George Leitmann

    1. University of California, Berkeley, USA

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Part of the book series: Mathematical Concepts and Methods in Science and Engineering (MCSENG, volume 24)

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Table of contents (17 chapters)

  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

      • George Leitmann
      Pages 7-24

    4. Integration of the Euler—Lagrange Equation

      • George Leitmann
      Pages 25-37

    5. An Inverse Problem

      • George Leitmann
      Pages 39-45

    6. The Weierstrass Necessary Condition

      • George Leitmann
      Pages 47-53

    7. Jacobi’s Necessary Condition

      • George Leitmann
      Pages 55-65

    8. Corner Conditions

      • George Leitmann
      Pages 67-70

    9. Concluding Remarks

      • George Leitmann
      Pages 71-73

  3. Optimal Control

    1. Front Matter

      Pages 75-75
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    2. Introduction

      • George Leitmann
      Pages 77-78

    3. Problem Statement and Optimality

      • George Leitmann
      Pages 79-98

    4. Regular Optimal Trajectories

      • George Leitmann
      Pages 99-123

    5. Examples of Extremal Control

      • George Leitmann
      Pages 125-138

    6. Some Generalizations

      • George Leitmann
      Pages 139-209

    7. Special Systems

      • George Leitmann
      Pages 211-239

    8. Sufficient Conditions

      • George Leitmann
      Pages 241-263

    9. Feedback Control

      • George Leitmann
      Pages 265-284

    10. Optimization with Vector — Valued Cost

      • George Leitmann
      Pages 285-299
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Keywords

About this book

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.

Authors and Affiliations

  • University of California, Berkeley, USA

    George Leitmann

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