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Calculus of Variations and Optimal Control
A. A. Milyutin, Russian Academy of Sciences, Moscow, Russia, and N. P. Osmolovskii, Moscow State Civil Engineering University, Russia

Translations of Mathematical Monographs
1998; 372 pp; hardcover
Volume: 180
ISBN-10: 0-8218-0753-6
ISBN-13: 978-0-8218-0753-8
List Price: US$135
Member Price: US$108
Order Code: MMONO/180
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The theory of a Pontryagin minimum is developed for problems in the calculus of variations. The application of the notion of a Pontryagin minimum to the calculus of variations is a distinctive feature of this book. A new theory of quadratic conditions for a Pontryagin minimum, which covers broken extremals, is developed, and corresponding sufficient conditions for a strong minimum are obtained. Some classical theorems of the calculus of variations are generalized.


Graduate students and research mathematicians working in the calculus of variations and control theory and in applications to mechanics, physics, and engineering.


"The first author is one of the patriarchs of the theory of optimal control who was most instrumental in introducing functional-theoretic methods into the theory at an early stage of its development. This is a highly original treatise full of new results and ideas which should be recommended to all interested in the theory of necessary and sufficient conditions in the calculus of variations and optimal control: strongly recommended to specialists and users ... [T]he collection of examples and applications in the book seems to be unrivaled in the literature, as regards both the number and the thoroughness of the analyses. [T]he book is certainly at the very forefront of the developments and may, in many respects, determine the state-of-the-art and directions for future studies in the areas to which it relates."

-- Mathematical Reviews

"The book is accessible to graduate students in mathematics and can be recommended to all of those who use extremum theory in their research or applied study."

-- Zentralblatt MATH

Table of Contents

First order conditions
  • First order conditions
  • Theory of a weak minimum for the problem on a fixed time interval
  • Theory of the maximum principle
  • Extremals and the Hamiltonian of a control system
  • Hamilton-Jacobi equation and field theory
  • Transformations of problems and invariance of extremals
Quadratic conditions
  • Quadratic conditions and conjugate points for broken extremals
  • Quadratic conditions for a Pontryagin minimum and sufficient conditions for a strong minimum: Proofs
  • Quadratic conditions in the general problem of the calculus of variations and related optimal control problems
  • Investigation of extremals by quadratic conditions: Examples
  • Bibliography
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