Topics in Optimization

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English Pages [486] Year 1967

Author / Uploaded
Leitmann G. (ed.)

Table of contents :
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 b.(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