Connectedness and Necessary Conditions for an Extremum

  • Alexander P. Abramov

Part of the Mathematics and Its Applications book series (MAIA, volume 431)

Table of contents

  1. Front Matter
    Pages i-xi
  2. Alexander P. Abramov
    Pages 1-20
  3. Alexander P. Abramov
    Pages 157-188
  4. Back Matter
    Pages 189-203

About this book


The present book is the outcome of efforts to introduce topological connectedness as one of the basic tools for the study of necessary conditions for an extremum. Apparently this monograph is the first book in the theory of maxima and minima where topological connectedness is used so widely for this purpose. Its application permits us to obtain new results in this sphere and to consider the classical results from a nonstandard point of view. Regarding the style of the present book it should be remarked that it is comparatively elementary. The author has made constant efforts to make the book as self-contained as possible. Certainly, familiarity with the basic facts of topology, functional analysis, and the theory of optimization is assumed. The book is written for applied mathematicians and graduate students interested in the theory of optimization and its applications. We present the synthesis of the well known Dybovitskii'-Milyutin ap­ proach for the study of necessary conditions for an extremum, based on functional analysis, and topological methods. This synthesis allows us to show that in some cases we have the following important result: if the Euler equation has no non trivial solution at a point of an extremum, then some inclusion is valid for the functionals belonging to the dual space. This general result is obtained for an optimization problem considered in a lin­ ear topological space. We also show an application of our result to some problems of nonlinear programming and optimal control.


Euler–Lagrange equation Optimal control functional analysis linear optimization nonlinear optimization optimization

Authors and affiliations

  • Alexander P. Abramov
    • 1
  1. 1.Computing CentreRussian Academy of SciencesMoscowRussia

Bibliographic information

  • DOI
  • Copyright Information Springer Science+Business Media B.V. 1998
  • Publisher Name Springer, Dordrecht
  • eBook Packages Springer Book Archive
  • Print ISBN 978-90-481-4981-0
  • Online ISBN 978-94-015-9119-5
  • Buy this book on publisher's site
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