Lectures on Mathematical Theory of Extremum Problems

1. Front Matter

   Pages i-v

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2. Editor’s Preface

   • Igor Vladimirovich Girsanov, B. T. Poljak

   Pages 1-1

3. Introduction

   • Igor Vladimirovich Girsanov, B. T. Poljak

   Pages 2-10

4. Topological Linear Spaces, Convex Sets, Weak Topologies

   • Igor Vladimirovich Girsanov, B. T. Poljak

   Pages 11-20

5. Hahn-Banach Theorem

   • Igor Vladimirovich Girsanov, B. T. Poljak

   Pages 21-24

6. Supporting Hyperplanes and Extremal Points

   • Igor Vladimirovich Girsanov, B. T. Poljak

   Pages 25-29

7. Cones, Dual Cones

   • Igor Vladimirovich Girsanov, B. T. Poljak

   Pages 30-37

8. Necessary Extremum Conditions (Euler-Lagrange Equation)

   • Igor Vladimirovich Girsanov, B. T. Poljak

   Pages 38-42

9. Directions of Decrease

   • Igor Vladimirovich Girsanov, B. T. Poljak

   Pages 43-57

10. Feasible Directions

    • Igor Vladimirovich Girsanov, B. T. Poljak

    Pages 58-60

11. Tangent Directions

    • Igor Vladimirovich Girsanov, B. T. Poljak

    Pages 61-68

12. Calculation of Dual Cones

    • Igor Vladimirovich Girsanov, B. T. Poljak

    Pages 69-77

13. Lagrange Multipliers and the Kuhn-Tucker Theorem

    • Igor Vladimirovich Girsanov, B. T. Poljak

    Pages 78-82

14. Problem of Optimal Control. Local Maximum Principle

    • Igor Vladimirovich Girsanov, B. T. Poljak

    Pages 83-92

15. Problem of Optimal Control. Maximum Principle

    • Igor Vladimirovich Girsanov, B. T. Poljak

    Pages 93-104

16. Problem of Optimal Control. Constraints on Phase Coordinates, Minimax Problem

    • Igor Vladimirovich Girsanov, B. T. Poljak

    Pages 105-113

17. Sufficient Extremum Conditions

    • Igor Vladimirovich Girsanov, B. T. Poljak

    Pages 114-120

18. Sufficient Extremum Conditions. Examples

    • Igor Vladimirovich Girsanov, B. T. Poljak

    Pages 121-123

19. Back Matter

    Pages 124-137

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The author of this book, Igor' Vladimirovich Girsanov, was one of the first mathematicians to study general extremum problems and to realize the feasibility and desirability of a unified theory of extremal problems, based on a functional­ analytic approach. He actively advocated this view, and his special course, given at the Faculty of Mechanics and Mathematics of the Moscow State University in 1963 and 1964, was apparently the first systematic exposition of a unified approach to the theory of extremal problems. This approach was based on the ideas of Dubovitskii and Milyutin [1]. The general theory of extremal problems has developed so intensely during the past few years that its basic concepts may now be considered finalized. Nevertheless, as yet the basic results of this new field of mathematics have not been presented in a form accessible to a wide range of readers. (The profound paper of Dubovitskii and Milyutin [2] can hardly be recommended for a first study of the theory, since, in particular, it does not contain proofs of the fundamental theorems. ) Girsanov's book fills this gap. It contains a systematic exposition of the general principles underlying the derivation of necessary and sufficient conditions for an extremum, in a wide variety of problems. Numerous applications are given to specific extremal problems. The main material is preceded by an introductory section in which all prerequisites from functional analysis are presented.

• Calculation
• Extremum
• Lie
• Topology
• equation
• extrema
• functional analysis
• maximum
• minimum
• theorem

• Computer Center, Moscow State University, Moscow, USSR

  B. T. Poljak

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