**Author**: Daniel Léonard

**Publisher:**Cambridge University Press

**ISBN:**9780521337465

**Category :**Business & Economics

**Languages :**en

**Pages :**372

Skip to content
## Bungei Journalism

# Optimal Control Theory and Static Optimization in Economics PDF Download

Are you looking for read ebook online? Search for your book and save it on your Kindle device, PC, phones or tablets. **Download Optimal Control Theory and Static Optimization in Economics PDF full book**. Access full book title **Optimal Control Theory and Static Optimization in Economics** by Daniel Léonard. Download full books in PDF and EPUB format.
## Optimal Control Theory and Static Optimization in Economics

**Author**: Daniel Léonard

**Publisher:** Cambridge University Press

**ISBN:** 9780521337465

**Category : **Business & Economics

**Languages : **en

**Pages : **372

**Book Description**

Optimal control theory is a technique being used increasingly by academic economists to study problems involving optimal decisions in a multi-period framework. This textbook is designed to make the difficult subject of optimal control theory easily accessible to economists while at the same time maintaining rigour. Economic intuitions are emphasized, and examples and problem sets covering a wide range of applications in economics are provided to assist in the learning process. Theorems are clearly stated and their proofs are carefully explained. The development of the text is gradual and fully integrated, beginning with simple formulations and progressing to advanced topics such as control parameters, jumps in state variables, and bounded state space. For greater economy and elegance, optimal control theory is introduced directly, without recourse to the calculus of variations. The connection with the latter and with dynamic programming is explained in a separate chapter. A second purpose of the book is to draw the parallel between optimal control theory and static optimization. Chapter 1 provides an extensive treatment of constrained and unconstrained maximization, with emphasis on economic insight and applications. Starting from basic concepts, it derives and explains important results, including the envelope theorem and the method of comparative statics. This chapter may be used for a course in static optimization. The book is largely self-contained. No previous knowledge of differential equations is required.

## Optimal Control Theory

**Author**: Suresh P. Sethi

**Publisher:** Springer Nature

**ISBN:** 3030917452

**Category : **Business & Economics

**Languages : **en

**Pages : **506

**Book Description**

This new 4th edition offers an introduction to optimal control theory and its diverse applications in management science and economics. It introduces students to the concept of the maximum principle in continuous (as well as discrete) time by combining dynamic programming and Kuhn-Tucker theory. While some mathematical background is needed, the emphasis of the book is not on mathematical rigor, but on modeling realistic situations encountered in business and economics. It applies optimal control theory to the functional areas of management including finance, production and marketing, as well as the economics of growth and of natural resources. In addition, it features material on stochastic Nash and Stackelberg differential games and an adverse selection model in the principal-agent framework. Exercises are included in each chapter, while the answers to selected exercises help deepen readers’ understanding of the material covered. Also included are appendices of supplementary material on the solution of differential equations, the calculus of variations and its ties to the maximum principle, and special topics including the Kalman filter, certainty equivalence, singular control, a global saddle point theorem, Sethi-Skiba points, and distributed parameter systems. Optimal control methods are used to determine optimal ways to control a dynamic system. The theoretical work in this field serves as the foundation for the book, in which the author applies it to business management problems developed from his own research and classroom instruction. The new edition has been refined and updated, making it a valuable resource for graduate courses on applied optimal control theory, but also for financial and industrial engineers, economists, and operational researchers interested in applying dynamic optimization in their fields.

## Optimal Control Theory and Static Optimization in Economics

**Author**: Daniel Léonard

**Publisher:** Cambridge University Press

**ISBN:** 9780521337465

**Category : **Business & Economics

**Languages : **en

**Pages : **372

**Book Description**

Optimal control theory is a technique being used increasingly by academic economists to study problems involving optimal decisions in a multi-period framework. This textbook is designed to make the difficult subject of optimal control theory easily accessible to economists while at the same time maintaining rigour. Economic intuitions are emphasized, and examples and problem sets covering a wide range of applications in economics are provided to assist in the learning process. Theorems are clearly stated and their proofs are carefully explained. The development of the text is gradual and fully integrated, beginning with simple formulations and progressing to advanced topics such as control parameters, jumps in state variables, and bounded state space. For greater economy and elegance, optimal control theory is introduced directly, without recourse to the calculus of variations. The connection with the latter and with dynamic programming is explained in a separate chapter. A second purpose of the book is to draw the parallel between optimal control theory and static optimization. Chapter 1 provides an extensive treatment of constrained and unconstrained maximization, with emphasis on economic insight and applications. Starting from basic concepts, it derives and explains important results, including the envelope theorem and the method of comparative statics. This chapter may be used for a course in static optimization. The book is largely self-contained. No previous knowledge of differential equations is required.

## Optimal Control Theory

**Author**: Donald E. Kirk

**Publisher:** Courier Corporation

**ISBN:** 9780486434841

**Category : **Technology & Engineering

**Languages : **en

**Pages : **484

**Book Description**

Geared toward upper-level undergraduates, this text introduces three aspects of optimal control theory: dynamic programming, Pontryagin's minimum principle, and numerical techniques for trajectory optimization. Numerous problems, which introduce additional topics and illustrate basic concepts, appear throughout the text. Solution guide available upon request. 131 figures. 14 tables. 1970 edition.

## Application of Optimal Control Theory to Enhanced Oil Recovery

**Author**: W. Fred Ramirez

**Publisher:** Elsevier

**ISBN:** 9780080868790

**Category : **Technology & Engineering

**Languages : **en

**Pages : **242

**Book Description**

In recent years, enhanced oil recovery techniques have received much attention in the oil industry. Enhanced oil recovery methods can be divided into three major categories: thermal processes which include steam flooding, steam stimulation, and in-situ combustion; chemical processes which include surfactant-polymer injection, polymer flooding, and caustic flooding; and miscible displacement processes which include miscible hydrocarbon displacement, carbon dioxide injection of large amounts of rather expensive fluids into oil bearing reservoir formations. Commercial application of any enhanced oil recovery process relies upon economic projections that show a decent return on the investment. Because of high chemical costs, it is important to optimize enhanced oil recovery processes to provide the greatest recovery at the lowest chemical injection cost. The aim of this book is to develop an optimal control theory for the determination of operating strategies that maximize the economic attractiveness of enhanced oil recovery processes. The determination of optimal control histories or operating strategies is one of the key elements in the successful usage of new enhanced oil recovery techniques. The information contained in the book will therefore be both interesting and useful to all those working in petroleum engineering, petroleum management and chemical engineering.

## Optimal Control Theory

**Author**: Suresh P. Sethi

**Publisher:** Springer

**ISBN:** 3319982370

**Category : **Business & Economics

**Languages : **en

**Pages : **565

**Book Description**

This fully revised 3rd edition offers an introduction to optimal control theory and its diverse applications in management science and economics. It brings to students the concept of the maximum principle in continuous, as well as discrete, time by using dynamic programming and Kuhn-Tucker theory. While some mathematical background is needed, the emphasis of the book is not on mathematical rigor, but on modeling realistic situations faced in business and economics. The book exploits optimal control theory to the functional areas of management including finance, production and marketing and to economics of growth and of natural resources. In addition, this new edition features materials on stochastic Nash and Stackelberg differential games and an adverse selection model in the principal-agent framework. The book provides exercises for each chapter and answers to selected exercises to help deepen the understanding of the material presented. Also included are appendices comprised of supplementary material on the solution of differential equations, the calculus of variations and its relationships to the maximum principle, and special topics including the Kalman filter, certainty equivalence, singular control, a global saddle point theorem, Sethi-Skiba points, and distributed parameter systems. Optimal control methods are used to determine optimal ways to control a dynamic system. The theoretical work in this field serves as a foundation for the book, which the author has applied to business management problems developed from his research and classroom instruction. The new edition has been completely refined and brought up to date. Ultimately this should continue to be a valuable resource for graduate courses on applied optimal control theory, but also for financial and industrial engineers, economists, and operational researchers concerned with the application of dynamic optimization in their fields.

## An Introduction to Optimal Control Theory

**Author**: Aaron Strauss

**Publisher:** Springer Science & Business Media

**ISBN:** 3642510019

**Category : **Business & Economics

**Languages : **en

**Pages : **154

**Book Description**

This paper is intended for the beginner. It is not a state of-the-art paper for research workers in the field of control theory. Its purpose is to introduce the reader to some of the problems and results in control theory, to illustrate the application of these re sults, and to provide a guide for his further reading on this subject. I have tried to motivate the results with examples, especial ly with one canonical, simple example described in §3. Many results, such as the maximum principle, have long and difficult proofs. I have omitted these proofs. In general I have included only the proofs which are either (1) not too difficult or (2) fairly enlightening as to the nature of the result. I have, however, usually attempted to draw the strongest conclusion from a given proof. For example, many existing proofs in control theory for compact targets and uniqueness of solutions also hold for closed targets and non-uniqueness. Finally, at the end of each section I have given references to generalizations and origins of the results discussed in that section. I make no claim of completeness in the references, however, as I have often been content merely to refer the reader either to an exposition or to a paper which has an extensive bibliography. IV These 1ecture notes are revisions of notes I used for aseries of nine 1ectures on contro1 theory at the International Summer Schoo1 on Mathematica1 Systems and Economics held in Varenna, Ita1y, June 1967.

## Optimal Control Theory and Static Optimization in Economics

**Author**: Daniel Léonard

**Publisher:** Cambridge University Press

**ISBN:** 1107717175

**Category : **Business & Economics

**Languages : **en

**Pages : **368

**Book Description**

Optimal control theory is a technique being used increasingly by academic economists to study problems involving optimal decisions in a multi-period framework. This textbook is designed to make the difficult subject of optimal control theory easily accessible to economists while at the same time maintaining rigour. Economic intuitions are emphasized, and examples and problem sets covering a wide range of applications in economics are provided to assist in the learning process. Theorems are clearly stated and their proofs are carefully explained. The development of the text is gradual and fully integrated, beginning with simple formulations and progressing to advanced topics such as control parameters, jumps in state variables, and bounded state space. For greater economy and elegance, optimal control theory is introduced directly, without recourse to the calculus of variations. The connection with the latter and with dynamic programming is explained in a separate chapter. A second purpose of the book is to draw the parallel between optimal control theory and static optimization. Chapter 1 provides an extensive treatment of constrained and unconstrained maximization, with emphasis on economic insight and applications. Starting from basic concepts, it derives and explains important results, including the envelope theorem and the method of comparative statics. This chapter may be used for a course in static optimization. The book is largely self-contained. No previous knowledge of differential equations is required.

## A Primer on the Calculus of Variations and Optimal Control Theory

**Author**: Mike Mesterton-Gibbons

**Publisher:** American Mathematical Soc.

**ISBN:** 0821847724

**Category : **Calculus of variations

**Languages : **en

**Pages : **274

**Book Description**

The calculus of variations is used to find functions that optimize quantities expressed in terms of integrals. Optimal control theory seeks to find functions that minimize cost integrals for systems described by differential equations. This book is an introduction to both the classical theory of the calculus of variations and the more modern developments of optimal control theory from the perspective of an applied mathematician. It focuses on understanding concepts and how to apply them. The range of potential applications is broad: the calculus of variations and optimal control theory have been widely used in numerous ways in biology, criminology, economics, engineering, finance, management science, and physics. Applications described in this book include cancer chemotherapy, navigational control, and renewable resource harvesting. The prerequisites for the book are modest: the standard calculus sequence, a first course on ordinary differential equations, and some facility with the use of mathematical software. It is suitable for an undergraduate or beginning graduate course, or for self study. It provides excellent preparation for more advanced books and courses on the calculus of variations and optimal control theory.

## Optimal Control Theory and its Applications

**Author**: B. J. Kirby

**Publisher:** Springer Science & Business Media

**ISBN:** 3642482902

**Category : **Mathematics

**Languages : **en

**Pages : **404

**Book Description**

This work (in two parts), Lecture Notes in Economics and Mathe matical Systems, Volume 105 and 106, constitutes the Proceedings of the Fourteenth Biennual Seminar of the Canadian Mathematical Congress, which was held from August 12 to August 25, 1973 at the University of Western Ontario, London, Ontario. The Canadian Mathematical Congress has held Biennual Seminars since 19~7, and these have covered a wide range of topics. The Seminar reported in this publication was concerned with "Optimal Control Theory and its Applications", a subject chosen for its active ~rowth and its wide implications for other fields. Both these aspects are exemplified in these Proceedings. Some lectures provided excellent surveys of particular fields whereas others concentrated on the presentation of new results. There were six distinguished Principal Lecturers: H.T. Banks, A.R. Dobell, H. Halkin, J.L. Lions, R.M. Thrall and W.M. Wonham, all of whom gave five to ten lectures during the two weeks of the Seminar. Except for Dr. Dobell's, these will all be found in Volume 105. Besides the Principal Lecturers there were three Guest Lecturers: M.C. Delfour, V. Jurdjevic and S.P. Sethi, who presented substantial bodies of material in two or three lectures and which are included in Volume 106. Many of the participants also spoke and reports of ~0st of these have also been included (Volume 106).

## Principles of Optimal Control Theory

**Author**: R. Gamkrelidze

**Publisher:** Springer Science & Business Media

**ISBN:** 1468473980

**Category : **Science

**Languages : **en

**Pages : **175

**Book Description**

In the late 1950's, the group of Soviet mathematicians consisting of L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko made fundamental contributions to optimal control theory. Much of their work was collected in their monograph, The Mathematical Theory of Optimal Processes. Subsequently, Professor Gamkrelidze made further important contributions to the theory of necessary conditions for problems of optimal control and general optimization problems. In the present monograph, Professor Gamkrelidze presents his current view of the fundamentals of optimal control theory. It is intended for use in a one-semester graduate course or advanced undergraduate course. We are now making these ideas available in English to all those interested in optimal control theory. West Lafayette, Indiana, USA Leonard D. Berkovitz Translation Editor Vll Preface This book is based on lectures I gave at the Tbilisi State University during the fall of 1974. It contains, in essence, the principles of general control theory and proofs of the maximum principle and basic existence theorems of optimal control theory. Although the proofs of the basic theorems presented here are far from being the shortest, I think they are fully justified from the conceptual view point. In any case, the notions we introduce and the methods developed have one unquestionable advantage -they are constantly used throughout control theory, and not only for the proofs of the theorems presented in this book.

eBook Journalism in PDF, ePub, Mobi and Kindle

Optimal control theory is a technique being used increasingly by academic economists to study problems involving optimal decisions in a multi-period framework. This textbook is designed to make the difficult subject of optimal control theory easily accessible to economists while at the same time maintaining rigour. Economic intuitions are emphasized, and examples and problem sets covering a wide range of applications in economics are provided to assist in the learning process. Theorems are clearly stated and their proofs are carefully explained. The development of the text is gradual and fully integrated, beginning with simple formulations and progressing to advanced topics such as control parameters, jumps in state variables, and bounded state space. For greater economy and elegance, optimal control theory is introduced directly, without recourse to the calculus of variations. The connection with the latter and with dynamic programming is explained in a separate chapter. A second purpose of the book is to draw the parallel between optimal control theory and static optimization. Chapter 1 provides an extensive treatment of constrained and unconstrained maximization, with emphasis on economic insight and applications. Starting from basic concepts, it derives and explains important results, including the envelope theorem and the method of comparative statics. This chapter may be used for a course in static optimization. The book is largely self-contained. No previous knowledge of differential equations is required.

This new 4th edition offers an introduction to optimal control theory and its diverse applications in management science and economics. It introduces students to the concept of the maximum principle in continuous (as well as discrete) time by combining dynamic programming and Kuhn-Tucker theory. While some mathematical background is needed, the emphasis of the book is not on mathematical rigor, but on modeling realistic situations encountered in business and economics. It applies optimal control theory to the functional areas of management including finance, production and marketing, as well as the economics of growth and of natural resources. In addition, it features material on stochastic Nash and Stackelberg differential games and an adverse selection model in the principal-agent framework. Exercises are included in each chapter, while the answers to selected exercises help deepen readers’ understanding of the material covered. Also included are appendices of supplementary material on the solution of differential equations, the calculus of variations and its ties to the maximum principle, and special topics including the Kalman filter, certainty equivalence, singular control, a global saddle point theorem, Sethi-Skiba points, and distributed parameter systems. Optimal control methods are used to determine optimal ways to control a dynamic system. The theoretical work in this field serves as the foundation for the book, in which the author applies it to business management problems developed from his own research and classroom instruction. The new edition has been refined and updated, making it a valuable resource for graduate courses on applied optimal control theory, but also for financial and industrial engineers, economists, and operational researchers interested in applying dynamic optimization in their fields.

Optimal control theory is a technique being used increasingly by academic economists to study problems involving optimal decisions in a multi-period framework. This textbook is designed to make the difficult subject of optimal control theory easily accessible to economists while at the same time maintaining rigour. Economic intuitions are emphasized, and examples and problem sets covering a wide range of applications in economics are provided to assist in the learning process. Theorems are clearly stated and their proofs are carefully explained. The development of the text is gradual and fully integrated, beginning with simple formulations and progressing to advanced topics such as control parameters, jumps in state variables, and bounded state space. For greater economy and elegance, optimal control theory is introduced directly, without recourse to the calculus of variations. The connection with the latter and with dynamic programming is explained in a separate chapter. A second purpose of the book is to draw the parallel between optimal control theory and static optimization. Chapter 1 provides an extensive treatment of constrained and unconstrained maximization, with emphasis on economic insight and applications. Starting from basic concepts, it derives and explains important results, including the envelope theorem and the method of comparative statics. This chapter may be used for a course in static optimization. The book is largely self-contained. No previous knowledge of differential equations is required.

Geared toward upper-level undergraduates, this text introduces three aspects of optimal control theory: dynamic programming, Pontryagin's minimum principle, and numerical techniques for trajectory optimization. Numerous problems, which introduce additional topics and illustrate basic concepts, appear throughout the text. Solution guide available upon request. 131 figures. 14 tables. 1970 edition.

In recent years, enhanced oil recovery techniques have received much attention in the oil industry. Enhanced oil recovery methods can be divided into three major categories: thermal processes which include steam flooding, steam stimulation, and in-situ combustion; chemical processes which include surfactant-polymer injection, polymer flooding, and caustic flooding; and miscible displacement processes which include miscible hydrocarbon displacement, carbon dioxide injection of large amounts of rather expensive fluids into oil bearing reservoir formations. Commercial application of any enhanced oil recovery process relies upon economic projections that show a decent return on the investment. Because of high chemical costs, it is important to optimize enhanced oil recovery processes to provide the greatest recovery at the lowest chemical injection cost. The aim of this book is to develop an optimal control theory for the determination of operating strategies that maximize the economic attractiveness of enhanced oil recovery processes. The determination of optimal control histories or operating strategies is one of the key elements in the successful usage of new enhanced oil recovery techniques. The information contained in the book will therefore be both interesting and useful to all those working in petroleum engineering, petroleum management and chemical engineering.

This fully revised 3rd edition offers an introduction to optimal control theory and its diverse applications in management science and economics. It brings to students the concept of the maximum principle in continuous, as well as discrete, time by using dynamic programming and Kuhn-Tucker theory. While some mathematical background is needed, the emphasis of the book is not on mathematical rigor, but on modeling realistic situations faced in business and economics. The book exploits optimal control theory to the functional areas of management including finance, production and marketing and to economics of growth and of natural resources. In addition, this new edition features materials on stochastic Nash and Stackelberg differential games and an adverse selection model in the principal-agent framework. The book provides exercises for each chapter and answers to selected exercises to help deepen the understanding of the material presented. Also included are appendices comprised of supplementary material on the solution of differential equations, the calculus of variations and its relationships to the maximum principle, and special topics including the Kalman filter, certainty equivalence, singular control, a global saddle point theorem, Sethi-Skiba points, and distributed parameter systems. Optimal control methods are used to determine optimal ways to control a dynamic system. The theoretical work in this field serves as a foundation for the book, which the author has applied to business management problems developed from his research and classroom instruction. The new edition has been completely refined and brought up to date. Ultimately this should continue to be a valuable resource for graduate courses on applied optimal control theory, but also for financial and industrial engineers, economists, and operational researchers concerned with the application of dynamic optimization in their fields.

This paper is intended for the beginner. It is not a state of-the-art paper for research workers in the field of control theory. Its purpose is to introduce the reader to some of the problems and results in control theory, to illustrate the application of these re sults, and to provide a guide for his further reading on this subject. I have tried to motivate the results with examples, especial ly with one canonical, simple example described in §3. Many results, such as the maximum principle, have long and difficult proofs. I have omitted these proofs. In general I have included only the proofs which are either (1) not too difficult or (2) fairly enlightening as to the nature of the result. I have, however, usually attempted to draw the strongest conclusion from a given proof. For example, many existing proofs in control theory for compact targets and uniqueness of solutions also hold for closed targets and non-uniqueness. Finally, at the end of each section I have given references to generalizations and origins of the results discussed in that section. I make no claim of completeness in the references, however, as I have often been content merely to refer the reader either to an exposition or to a paper which has an extensive bibliography. IV These 1ecture notes are revisions of notes I used for aseries of nine 1ectures on contro1 theory at the International Summer Schoo1 on Mathematica1 Systems and Economics held in Varenna, Ita1y, June 1967.

Optimal control theory is a technique being used increasingly by academic economists to study problems involving optimal decisions in a multi-period framework. This textbook is designed to make the difficult subject of optimal control theory easily accessible to economists while at the same time maintaining rigour. Economic intuitions are emphasized, and examples and problem sets covering a wide range of applications in economics are provided to assist in the learning process. Theorems are clearly stated and their proofs are carefully explained. The development of the text is gradual and fully integrated, beginning with simple formulations and progressing to advanced topics such as control parameters, jumps in state variables, and bounded state space. For greater economy and elegance, optimal control theory is introduced directly, without recourse to the calculus of variations. The connection with the latter and with dynamic programming is explained in a separate chapter. A second purpose of the book is to draw the parallel between optimal control theory and static optimization. Chapter 1 provides an extensive treatment of constrained and unconstrained maximization, with emphasis on economic insight and applications. Starting from basic concepts, it derives and explains important results, including the envelope theorem and the method of comparative statics. This chapter may be used for a course in static optimization. The book is largely self-contained. No previous knowledge of differential equations is required.

The calculus of variations is used to find functions that optimize quantities expressed in terms of integrals. Optimal control theory seeks to find functions that minimize cost integrals for systems described by differential equations. This book is an introduction to both the classical theory of the calculus of variations and the more modern developments of optimal control theory from the perspective of an applied mathematician. It focuses on understanding concepts and how to apply them. The range of potential applications is broad: the calculus of variations and optimal control theory have been widely used in numerous ways in biology, criminology, economics, engineering, finance, management science, and physics. Applications described in this book include cancer chemotherapy, navigational control, and renewable resource harvesting. The prerequisites for the book are modest: the standard calculus sequence, a first course on ordinary differential equations, and some facility with the use of mathematical software. It is suitable for an undergraduate or beginning graduate course, or for self study. It provides excellent preparation for more advanced books and courses on the calculus of variations and optimal control theory.

This work (in two parts), Lecture Notes in Economics and Mathe matical Systems, Volume 105 and 106, constitutes the Proceedings of the Fourteenth Biennual Seminar of the Canadian Mathematical Congress, which was held from August 12 to August 25, 1973 at the University of Western Ontario, London, Ontario. The Canadian Mathematical Congress has held Biennual Seminars since 19~7, and these have covered a wide range of topics. The Seminar reported in this publication was concerned with "Optimal Control Theory and its Applications", a subject chosen for its active ~rowth and its wide implications for other fields. Both these aspects are exemplified in these Proceedings. Some lectures provided excellent surveys of particular fields whereas others concentrated on the presentation of new results. There were six distinguished Principal Lecturers: H.T. Banks, A.R. Dobell, H. Halkin, J.L. Lions, R.M. Thrall and W.M. Wonham, all of whom gave five to ten lectures during the two weeks of the Seminar. Except for Dr. Dobell's, these will all be found in Volume 105. Besides the Principal Lecturers there were three Guest Lecturers: M.C. Delfour, V. Jurdjevic and S.P. Sethi, who presented substantial bodies of material in two or three lectures and which are included in Volume 106. Many of the participants also spoke and reports of ~0st of these have also been included (Volume 106).

In the late 1950's, the group of Soviet mathematicians consisting of L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko made fundamental contributions to optimal control theory. Much of their work was collected in their monograph, The Mathematical Theory of Optimal Processes. Subsequently, Professor Gamkrelidze made further important contributions to the theory of necessary conditions for problems of optimal control and general optimization problems. In the present monograph, Professor Gamkrelidze presents his current view of the fundamentals of optimal control theory. It is intended for use in a one-semester graduate course or advanced undergraduate course. We are now making these ideas available in English to all those interested in optimal control theory. West Lafayette, Indiana, USA Leonard D. Berkovitz Translation Editor Vll Preface This book is based on lectures I gave at the Tbilisi State University during the fall of 1974. It contains, in essence, the principles of general control theory and proofs of the maximum principle and basic existence theorems of optimal control theory. Although the proofs of the basic theorems presented here are far from being the shortest, I think they are fully justified from the conceptual view point. In any case, the notions we introduce and the methods developed have one unquestionable advantage -they are constantly used throughout control theory, and not only for the proofs of the theorems presented in this book.