Topics in Optimization

List of ContributorsPrefacePart 1 A Variational Approach Chapter 1. Inequalities in a Variational Problem 1.0 Introduction 1.1 Condition 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 Ia, 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 of End Effects in an MHD Channel Appendix ReferencesPart 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.