**Translations of Mathematical Monographs**

1998;
372 pp;
Hardcover

MSC: Primary 49;

**Print ISBN: 978-0-8218-0753-8
Product Code: MMONO/180**

List Price: $135.00

AMS Member Price: $108.00

MAA Member Price: $121.50

**Electronic ISBN: 978-1-4704-4594-2
Product Code: MMONO/180.E**

List Price: $135.00

AMS Member Price: $108.00

MAA Member Price: $121.50

# Calculus of Variations and Optimal Control

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*A. A. Milyutin; N. P. Osmolovskii*

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.

#### Readership

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

#### Reviews & Endorsements

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

# Table of Contents

## Calculus of Variations and Optimal Control

- Cover Cover11
- Title page v6
- Contents vii8
- Preface xi12
- Introduction 114
- First order conditions 518
- Theory of a weak minimum for the problem on a fixed time interval 720
- Theory of the maximum principle 2336
- Extremals and the Hamiltonian of a control system 4356
- Hamilton-Jacobi equation and field theory 87100
- Transformations of problems and invariance of extremals 125138
- Quadratic conditions 153166
- Quadratic conditions and conjugate points for broken extremals 155168
- Quadratic conditions for a Pontryagin minimum and sufficient conditions for a strong minimum: Proofs 211224
- Quadratic conditions in the general problem of the calculus of variations and related optimal control problems 283296
- Investigation of extremals by quadratic conditions: Examples 325338
- Bibliography 371384
- Back Cover Back Cover1386