Topics in Optimization
Published on 14. May 2014
487 pages
978-0-08-095538-4 (ISBN)
Description
Topics in Optimization
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Series
Language
English
Place of publication
San Diego
United States
ISBN-13
978-0-08-095538-4 (9780080955384)
Copyright in bibliographic data and cover images is held by Nielsen Book Services Limited or by the publishers or by their respective licensors: all rights reserved.
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George Leitmann
Topics in Optimization
Book
04/1967
Academic Press
€90.37
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Content
- Front Cover
- TOPICS IN OPTIMIZATION
- Copyright Page
- Contents
- List of Contributors
- Preface
- Part 1 A Variational Approach
- Chapter 1. Inequalities in a Variational Problem
- 1.0 Introduction
- 1.1 Condition I
- 1.2 Conditions II and III
- PART A-CASE (a)
- 1.3 Preliminary Considerations
- 1.4 Singularities in Case (a)
- 1.5 The Extremaloid Index
- 1.6 The Imbedding Construction
- 1.7 Condition IV
- 1.8 Proof of Sufficiency
- 1.9 Numerical Example
- PART B-CASE (b)
- 1.10 Preliminary Considerations
- 1.11 Singularities in Case (b)
- 1.12 The Imbedding Construction
- 1.13 Numerical Example
- 1.14 Discussion of the Results
- References
- Chapter 2. Discontinuities in a Variational Problem
- 2.0 Introduction
- 2.1 Conditions Ic and Id
- PART A-CASE (a)
- 2.2 Conditions la, Ib, II, and III
- 2.3 Preliminary Considerations
- 2.4 The Function h(y')
- 2.5 Zermelo Diagram
- 2.6 The Imbedding Construction
- 2.7 Condition IV'
- 2.8 The Hilbert Integral
- 2.9 Proof of Sufficiency
- 2.10 Numerical Example
- 2.11 Discussion of the Results
- PART B-CASE (b)
- 2.12 Conditions Ia, Ib, II, and III
- 2.13 Preliminary Considerations
- 2.14 Zermelo Diagram
- 2.15 Corner Manifolds
- 2.16 Conditions II' and IIN'
- 2.17 Free Corners
- 2.18 A Special Case
- 2.19 The Imbedding Construction
- 2.20 Proof of Sufficiency
- 2.21 Numerical Example
- 2.22 Discussion of the Results
- References
- Chapter 3. Singular Extremals
- 3.0 Introduction
- 3.1 Second Variation Test for Singular Extremals
- 3.2 A Transformation Approach to the Analysis of Singular Subarcs
- 3.3 Examples
- References
- Chapter 4. Thrust Programming in a Central Gravitational Field
- 4.1 General Equations Governing the Motion of a Boosting Vehicle in a Central Gravitational Field
- 4.2 Integrals of the Basic System of Equations
- 4.3 Boundary Conditions: Various Types of Motion
- 4.4 Orbits on a Spherical Surface
- 4.5 Boosting Devices of Limited Propulsive Power
- 4.6 Singular Control Regimes
- References
- Chapter 5. The Mayer-Bolza Problem for Multiple Integrals: Some Optimum Problems for Elliptic Differential Equations Arising in Magnetohydrodynamics
- 5.0 Introduction
- 5.1 Optimum Problems for Partial Differential Equations: Necessary Conditions of Optimality
- 5.2 Optimum Problems in the Theory of Magnetohydrodynamical Channel Flow
- 5.3 Application to the Theory of MHD Power Generation: Minimization ofEnd Effects in an MHD Channel
- Appendix
- References
- Part 2 A Geometric Approach
- Chapter 6. Mathematical Foundations of System Optimization
- 6.0 Introduction
- 6.1 Dynamical Polysystem
- 6.2 Optimization Problem
- 6.3 The Principle of Optimal Evolution
- 6.4 Statement of the Maximum Principle
- 6.5 Proof of the Maximum Principle for an Elementary Dynamical Polysystem
- 6.6 Proof of the Maximum Principle for a Linear Dynamical Polysystem
- 6.7 Proof of the Maximum Principle for a General Dynamical Polysystem
- 6.8 Uniformly Continuous Dependence of Trajectories with Respect to Variations of the Control Functions
- 6.9 Some Uniform Estimates for the Approximation z(t
- u) of the Variational Trajectory y(t
- u)
- 6.10 Convexity of the Range of a Vector Integral over the Class A of Subsets of [0, 1]
- 6.11 Proof of the Fundamental Lemma
- 6.12 An Intuitive Approach to the Maximum Principle
- Appendix A. Some Results from the Theory of Ordinary Differential Equations
- Appendix B. The Geometry of Convex Sets
- References
- Chapter 7. On the Geometry of Optimal Processes
- 7.0 Introduction
- 7.1 Dynamical System
- 7.2 Augmented State Space and Trajectories
- 7.3 Limiting Surfaces and Optimal Isocost Surfaces
- 7.4 Some Properties of Optimal Isocost Surfaces
- 7.5 Some Global Properties of Limiting Surfaces
- 7.6 Some Local Properties of Limiting Surfaces
- 7.7 Some Properties of Local Cones
- 7.8 Tangent Cone be(x)
- 7.9 A Nice Limiting Surface
- 7.10 A Set of Admissible Rules
- 7.11 Velocity Vectors in Augmented State Space
- 7.12 Separability of Local Cones
- 7.13 Regular and Nonregular Points of a Limiting Surface
- 7.14 Some Properties of a Linear Transformation
- 7.15 Properties of Separable Local Cones
- 7.16 Attractive and Repulsive Subsets of a Limiting Surface
- 7.17 Regular Subset of a Limiting Surface
- 7.18 Antiregular Subset of a Limiting Surface
- 7.19 Symmetrical Subset of Local Cone g(x)
- 7.20 A Maximum Principle
- 7.21 Boundary Points of E*
- 7.22 Boundary and Interior Points of E*
- 7.23 Degenerated Case
- 7.24 Some Illustrative Examples
- Appendix
- Bibliography
- Chapter 8. The Pontryagin Maximum Principle
- 8.0 Introduction
- 8.1 The Extended Problem
- 8.2 The Control Lemma
- 8.3 The Controllability Theorem
- 8.4 The Maximum Principle (Part I)
- 8.5 The Maximum Principle (Part II)
- 8.6 The Bang-Bang Principle
- Reference
- Chapter 9. Synthesis of Optimal Controls
- 9.0 Introduction
- 9.1 Neustadt's Synthesis Method
- 9.2 Computational Considerations
- 9.3 Final Remarks
- References
- Chapter 10. The Calculus of Variations, Functional Analysis, and Optimal Control Problems
- 10.0 Introduction
- 10.1 The Problem of Mayer
- 10.2 Optimal Control Problems
- 10.3 Abstract Analysis, Basic Concepts
- 10.4 The Multiplier Rule in Abstract Analysis
- 10.5 Necessary Conditions for Optimal Controls
- 10.6 Variation of Endpoints and Initial Conditions
- 10.7 Examples of Optimal Control Problems
- 10.8 A Convergent Gradient Procedure
- 10.9 Computations Using the Convergent Gradient Procedure
- References
- Author Index
- Subject Index
