Variational Calculus and Optimal Control

Optimization with Elementary Convexity

  • John L. Troutman

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

Table of contents

  1. Front Matter
    Pages i-xv
  2. Review of Optimization in ℝd

    1. John L. Troutman
      Pages 1-9
  3. Basic Theory

    1. Front Matter
      Pages 11-11
    2. John L. Troutman
      Pages 13-35
    3. John L. Troutman
      Pages 36-52
    4. John L. Troutman
      Pages 53-96
    5. John L. Troutman
      Pages 97-102
    6. John L. Troutman
      Pages 103-144
    7. John L. Troutman
      Pages 145-193
  4. Advanced Topics

    1. Front Matter
      Pages 195-196
    2. John L. Troutman
      Pages 197-233
    3. John L. Troutman
      Pages 234-281
    4. John L. Troutman
      Pages 282-337
  5. Optimal Control

    1. Front Matter
      Pages 339-339
    2. John L. Troutman
      Pages 341-377
    3. John L. Troutman
      Pages 378-418
  6. Back Matter
    Pages 419-462

About this book

Introduction

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.

Keywords

Calculus Convexity Konvexe Funktion Optimal control Variationsrechnung linear optimization optimization

Authors and affiliations

  • John L. Troutman
    • 1
  1. 1.Department of MathematicsSyracuse UniversitySyracuseUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4612-0737-5
  • Copyright Information Springer-Verlag New York, Inc. 1996
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4612-6887-1
  • Online ISBN 978-1-4612-0737-5
  • Series Print ISSN 0172-6056
  • About this book