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Control Theory and Optimization I

Homogeneous Spaces and the Riccati Equation in the Calculus of Variations

  • Book
  • © 2000

Overview

Authors:
  1. M. I. Zelikin

    1. Department of Mathematics, MGU, Vorob’evy Gory, Moscow, Russia

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  • The only monograph on this topic

Part of the book series: Encyclopaedia of Mathematical Sciences (EMS, volume 86)

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Table of contents (10 chapters)

  1. Front Matter

    Pages I-XII
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  2. Introduction

    • M. I. Zelikin
    Pages 1-4

  3. Classical Calculus of Variations

    • M. I. Zelikin
    Pages 5-59

  4. Riccati Equation in the Classical Calculus of Variations

    • M. I. Zelikin
    Pages 60-79

  5. Lie Groups and Lie Algebras

    • M. I. Zelikin
    Pages 80-111

  6. Grassmann Manifolds

    • M. I. Zelikin
    Pages 112-129

  7. Matrix Double Ratio

    • M. I. Zelikin
    Pages 130-165

  8. Complex Riccati Equations

    • M. I. Zelikin
    Pages 166-207

  9. Higher-Dimensional Calculus of Variations

    • M. I. Zelikin
    Pages 208-253

  10. On the Quadratic System of Partial Differential Equations Related to the Minimization Problem for a Multiple Integral

    • M. I. Zelikin
    Pages 254-271

  11. Epilogue

    • M. I. Zelikin
    Pages 272-272

  12. Back Matter

    Pages 273-284
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Keywords

About this book

This book is devoted to the development of geometrie methods for studying and revealing geometrie aspects of the theory of differential equations with quadratie right-hand sides (Riccati-type equations), which are closely related to the calculus of variations and optimal control theory. The book contains the following three parts, to each of which aseparate book could be devoted: 1. the classieal calculus of variations and the geometrie theory of the Riccati equation (Chaps. 1-5), 2. complex Riccati equations as flows on Cartan-Siegel homogeneity da­ mains (Chap. 6), and 3. the minimization problem for multiple integrals and Riccati partial dif­ ferential equations (Chaps. 7 and 8). Chapters 1-4 are mainly auxiliary. To make the presentation complete and self-contained, I here review the standard facts (needed in what folIows) from the calculus of variations, Lie groups and algebras, and the geometry of Grass­ mann and Lagrange-Grassmann manifolds. When choosing these facts, I pre­ fer to present not the most general but the simplest assertions. Moreover, I try to organize the presentation so that it is not obscured by formal and technical details and, at the same time, is sufficiently precise. Other chapters contain my results concerning the matrix double ratio, com­ plex Riccati equations, and also the Riccati partial differential equation, whieh the minimization problem for a multiple integral. arises in The book is based on a course of lectures given in the Department of Me­ and Mathematics of Moscow State University during several years.

Reviews

".... This book was written by a master expositor and is required reading for anyone who is intersted in pursuing a serious study of the Riccati equation. The first four chapters should be required reading for every graduate student who is thinking about studying geometric or mathematical control theory. I do not know of a better overview of the matheamtics required to do modern geometric control theory. Every control theorist should have a well-worn copy of this book on his bookshelf. ... Zelikin has written a book that will be well read for many years...."

Siam Review, Vol. 43/1, March 2001

"... The text requires good background, but will be a useful reference."

Mathematika 2002, Issue 93-94

Authors and Affiliations

  • Department of Mathematics, MGU, Vorob’evy Gory, Moscow, Russia

    M. I. Zelikin

Bibliographic Information

  • Book Title: Control Theory and Optimization I

  • Book Subtitle: Homogeneous Spaces and the Riccati Equation in the Calculus of Variations

  • Authors: M. I. Zelikin

  • Series Title: Encyclopaedia of Mathematical Sciences

  • DOI: https://doi.org/10.1007/978-3-662-04136-9

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag Berlin Heidelberg 2000

  • Hardcover ISBN: 978-3-540-66741-4Published: 15 December 1999

  • Softcover ISBN: 978-3-642-08603-8Published: 05 December 2010

  • eBook ISBN: 978-3-662-04136-9Published: 14 March 2013

  • Series ISSN: 0938-0396

  • Edition Number: 1

  • Number of Pages: XII, 284

  • Additional Information: Original Russian edition published by Faktorial, Moscow, 1998

  • Topics: Differential Geometry, Calculus of Variations and Optimal Control; Optimization

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