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Variational Calculus and Optimal Control

Optimization with Elementary Convexity

  • Textbook
  • © 1996

Overview

Authors:
  1. John L. Troutman

    1. Department of Mathematics, Syracuse University, Syracuse, USA

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Part of the book series: Undergraduate Texts in Mathematics (UTM)

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Table of contents (12 chapters)

  1. Front Matter

    Pages i-xv
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  2. Review of Optimization in ℝd

    1. Review of Optimization in ℝd

      • John L. Troutman
      Pages 1-9

  3. Basic Theory

    1. Front Matter

      Pages 11-11
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    2. Standard Optimization Problems

      • John L. Troutman
      Pages 13-35

    3. Linear Spaces and Gâteaux Variations

      • John L. Troutman
      Pages 36-52

    4. Minimization of Convex Functions

      • John L. Troutman
      Pages 53-96

    5. The Lemmas of Lagrange and du Bois-Reymond

      • John L. Troutman
      Pages 97-102

    6. Local Extrema in Normed Linear Spaces

      • John L. Troutman
      Pages 103-144

    7. The Euler-Lagrange Equations

      • John L. Troutman
      Pages 145-193

  4. Advanced Topics

    1. Front Matter

      Pages 195-196
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    2. Piecewise C1 Extremal Functions

      • John L. Troutman
      Pages 197-233

    3. Variational Principles in Mechanics

      • John L. Troutman
      Pages 234-281

    4. Sufficient Conditions for a Minimum

      • John L. Troutman
      Pages 282-337

  5. Optimal Control

    1. Front Matter

      Pages 339-339
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    2. Control Problems and Sufficiency Considerations

      • John L. Troutman
      Pages 341-377

    3. Necessary Conditions for Optimality

      • John L. Troutman
      Pages 378-418

  6. Back Matter

    Pages 419-462
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Keywords

About this book

Although the calculus of variations has ancient origins in questions of Ar­ istotle and Zenodoros, its mathematical principles first emerged in the post­ calculus investigations of Newton, the Bernoullis, Euler, and Lagrange. Its results now supply fundamental tools of exploration to both mathematicians and those in the applied sciences. (Indeed, the macroscopic statements ob­ tained through variational principles may provide the only valid mathemati­ cal formulations of many physical laws. ) Because of its classical origins, variational calculus retains the spirit of natural philosophy common to most mathematical investigations prior to this century. The original applications, including the Bernoulli problem of finding the brachistochrone, require opti­ mizing (maximizing or minimizing) the mass, force, time, or energy of some physical system under various constraints. The solutions to these problems satisfy related differential equations discovered by Euler and Lagrange, and the variational principles of mechanics (especially that of Hamilton from the last century) show the importance of also considering solutions that just provide stationary behavior for some measure of performance of the system. However, many recent applications do involve optimization, in particular, those concerned with problems in optimal control. Optimal control is the rapidly expanding field developed during the last half-century to analyze optimal behavior of a constrained process that evolves in time according to prescribed laws. Its applications now embrace a variety of new disciplines, including economics and production planning.

Authors and Affiliations

  • Department of Mathematics, Syracuse University, Syracuse, USA

    John L. Troutman

Bibliographic Information

  • Book Title: Variational Calculus and Optimal Control

  • Book Subtitle: Optimization with Elementary Convexity

  • Authors: John L. Troutman

  • Series Title: Undergraduate Texts in Mathematics

  • DOI: https://doi.org/10.1007/978-1-4612-0737-5

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media New York 1996

  • Hardcover ISBN: 978-0-387-94511-8Published: 01 December 1995

  • Softcover ISBN: 978-1-4612-6887-1Published: 30 September 2012

  • eBook ISBN: 978-1-4612-0737-5Published: 06 December 2012

  • Series ISSN: 0172-6056

  • Series E-ISSN: 2197-5604

  • Edition Number: 2

  • Number of Pages: XV, 462

  • Topics: Calculus of Variations and Optimal Control; Optimization, Systems Theory, Control

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