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Optimization—Theory and Applications

Problems with Ordinary Differential Equations

  • Book
  • © 1983

Overview

Authors:
  1. Lamberto Cesari

    1. Department of Mathematics, University of Michigan, Ann Arbor, USA

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Part of the book series: Stochastic Modelling and Applied Probability (SMAP, volume 17)

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Table of contents (18 chapters)

  1. Front Matter

    Pages i-xv
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  2. Problems of Optimization—A General View

    • Lamberto Cesari
    Pages 1-23

  3. The Classical Problems of the Calculus of Variations: Necessary Conditions and Sufficient Conditions; Convexity and Lower Semicontinuity

    • Lamberto Cesari
    Pages 24-115

  4. Examples and Exercises on Classical Problems

    • Lamberto Cesari
    Pages 116-158

  5. Statement of the Necessary Condition for Mayer Problems of Optimal Control

    • Lamberto Cesari
    Pages 159-195

  6. Lagrange and Bolza Problems of Optimal Control and Other Problems

    • Lamberto Cesari
    Pages 196-205

  7. Examples and Exercises on Optimal Control

    • Lamberto Cesari
    Pages 206-232

  8. Proofs of the Necessary Condition for Control Problems and Related Topics

    • Lamberto Cesari
    Pages 233-270

  9. The Implicit Function Theorem and the Elementary Closure Theorem

    • Lamberto Cesari
    Pages 271-308

  10. Existence Theorems: The Bounded, or Elementary, Case

    • Lamberto Cesari
    Pages 309-324

  11. Closure and Lower Closure Theorems under Weak Convergence

    • Lamberto Cesari
    Pages 325-366

  12. Existence Theorems: Weak Convergence and Growth Conditions

    • Lamberto Cesari
    Pages 367-402

  13. Existence Theorems: The Case of an Exceptional Set of No Growth

    • Lamberto Cesari
    Pages 403-416

  14. Existence Theorems: The Use of Lipschitz and Tempered Growth Conditions

    • Lamberto Cesari
    Pages 417-429

  15. Existence Theorems: Problems of Slow Growth

    • Lamberto Cesari
    Pages 430-442

  16. Existence Theorems: The Use of Mere Pointwise Convergence on the Trajectories

    • Lamberto Cesari
    Pages 443-452

  17. Existence Theorems: Problems with No Convexity Assumptions

    • Lamberto Cesari
    Pages 453-473

  18. Duality and Upper Semicontinuity of Set Valued Functions

    • Lamberto Cesari
    Pages 474-502

  19. Approximation of Usual and of Generalized Solutions

    • Lamberto Cesari
    Pages 503-521

  20. Back Matter

    Pages 523-542
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Keywords

About this book

This book has grown out of lectures and courses in calculus of variations and optimization taught for many years at the University of Michigan to graduate students at various stages of their careers, and always to a mixed audience of students in mathematics and engineering. It attempts to present a balanced view of the subject, giving some emphasis to its connections with the classical theory and to a number of those problems of economics and engineering which have motivated so many of the present developments, as well as presenting aspects of the current theory, particularly value theory and existence theorems. However, the presentation ofthe theory is connected to and accompanied by many concrete problems of optimization, classical and modern, some more technical and some less so, some discussed in detail and some only sketched or proposed as exercises. No single part of the subject (such as the existence theorems, or the more traditional approach based on necessary conditions and on sufficient conditions, or the more recent one based on value function theory) can give a sufficient representation of the whole subject. This holds particularly for the existence theorems, some of which have been conceived to apply to certain large classes of problems of optimization. For all these reasons it is essential to present many examples (Chapters 3 and 6) before the existence theorems (Chapters 9 and 11-16), and to investigate these examples by means of the usual necessary conditions, sufficient conditions, and value function theory.

Authors and Affiliations

  • Department of Mathematics, University of Michigan, Ann Arbor, USA

    Lamberto Cesari

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