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1. Functionals

1.1 Introduction Examples of Optimizational Problems

1.2 Vector Spaces

1.3 Functionals

1.4 Normed Vector Spaces

1.5 Continuous Functionals

1.6 Linear Functionals

2. A Fundamental Necessary Condition for an Extremum

2.1 Introduction

2.2 A Fundamental Necessary Condition for an Extremum

2.3 Some Remarks on the Gâteaux Variation

2.4 Examples on the Calculation of Gâteaux Variations

2.5 An Optimization Problem in Production Planning

2.6 Some Remarks on the Frechet Differential

3. The Euler-Lagrange Necessary Condition for an Extremum with Constraints

3.1 Extremum Problems with a Single Constraint

3.2 Weak Continuity of Variations

3.3 Statement of the Euler-Lagrange Multiplier Theorem for a Single Constraint

3.4 Three Examples, and Some Remarks on the Geometrical Significance of the Multiplier Theorem

3.5 Proof of the Euler-Lagrange Multiplier Theorem

3.6 The Euler-Lagrange Multiplier Theorem for Many Constraints

3.7 An Optimum Consumption Policy with Terminal Savings Constraint During a Period of Inflation

3.8 The Meaning of the Euler-Lagrange Multipliers

3.9 Chaplygin's Problem, or a Modern Version of Queen Dido's Problem

3.10 The John Multiplier Theorem

4. Applications of the Euler-Lagrange Multiplier Theorem in the Calculus of Variations

4.1 Problems with Fixed End Points

4.2 John Bernoulli's Brachistochrone Problem, and Brachistochrones Through the Earth

4.3 Geodesic Curves

4.4 Problems with Variable End Points

4.5 How to Design a Thrilling Chute-the-Chute

4.6 Functionals Involving Several Unknown Functions

4.7 Fermat's Principle in Geometrical Optics

4.8 Hamilton's Principle of Stationary Action an Example on Small Vibrations

4.9 The McShane-Blankinship Curtain Rod Problem Functionals Involving Higher-Order Derivatives

4.10 Functionals Involving Several Independent Variables the Minimal Surface Problem

4.11 The Vibrating String

5. Applications of the Euler-Lagrange Multiplier Theorem to Problems with Global Pointwise Inequality Constraints

5.1 Slack Functions and Composite Curves

5.2 An Optimum Consumption Policy with Terminal Savings Constraint Without Extreme Hardship

5.3 A Problem in Production Planning with Inequality Constraints

6. Applications of the Euler-Lagrange Multiplier Theorem in Elementary Control Theory

6.1 Introduction

6.2 A Rocket Control Problem: Minimum Time

6.3 A Rocket Control Problem: Minimum Fuel

6.4 A More General Control Problem

6.5 A Simple Bang-Bang Problem

6.6 Some Remarks on the Maximum Principle and Dynamic Programming

7. The Variational Description of Sturm-Liouville Eigenvalues

7.1 Introduction to Sturm-Liouville Problems

7.2 The Relation Between the Lowest Eigenvalue and the Rayleigh Quotient

7.3 The Rayleigh-Ritz Method for the Lowest Eigenvalue

7.4 Higher Eigenvalues and the Rayleigh Quotient

7.5 The Courant Minimax Principle

7.6 Some Implications of the Courant Minimax Principle

7.7 Further Extensions of the Theory

7.8 Some General Remarks on the Ritz Method of Approximate Minimization

8. Some Remarks on the Use of the Second Variation in Extremum Problems

8.1 Higher-Order Variations

8.2 A Necessary Condition Involving the Second Variation at an Extremum

8.3 Sufficient Conditions for a Local Extremum

Appendix 1. The Cauchy and Schwarz Inequalities

Appendix 2. An Example on Normed Vector Spaces

Appendix 3. An Integral Inequality

Appendix 4. A Fundamental Lemma of the Calculus of Variations

Appendix 5. Du Bois-Reymond's Derivation of the Euler-Lagrange Equation

Appendix 6. A Useful Result from Calculus

Appendix 7. The Construction of a Certain Function

Appendix 8. The Fundamental Lemma for the Case of Several Independent Variables

Appendix 9. The Kinetic Energy for a Certain Model of an Elastic String

Appendix 10. The Variation of an Initial Value Problem with Respect to a Parameter

Subject Index Author Index

1.1 Introduction Examples of Optimizational Problems

1.2 Vector Spaces

1.3 Functionals

1.4 Normed Vector Spaces

1.5 Continuous Functionals

1.6 Linear Functionals

2. A Fundamental Necessary Condition for an Extremum

2.1 Introduction

2.2 A Fundamental Necessary Condition for an Extremum

2.3 Some Remarks on the Gâteaux Variation

2.4 Examples on the Calculation of Gâteaux Variations

2.5 An Optimization Problem in Production Planning

2.6 Some Remarks on the Frechet Differential

3. The Euler-Lagrange Necessary Condition for an Extremum with Constraints

3.1 Extremum Problems with a Single Constraint

3.2 Weak Continuity of Variations

3.3 Statement of the Euler-Lagrange Multiplier Theorem for a Single Constraint

3.4 Three Examples, and Some Remarks on the Geometrical Significance of the Multiplier Theorem

3.5 Proof of the Euler-Lagrange Multiplier Theorem

3.6 The Euler-Lagrange Multiplier Theorem for Many Constraints

3.7 An Optimum Consumption Policy with Terminal Savings Constraint During a Period of Inflation

3.8 The Meaning of the Euler-Lagrange Multipliers

3.9 Chaplygin's Problem, or a Modern Version of Queen Dido's Problem

3.10 The John Multiplier Theorem

4. Applications of the Euler-Lagrange Multiplier Theorem in the Calculus of Variations

4.1 Problems with Fixed End Points

4.2 John Bernoulli's Brachistochrone Problem, and Brachistochrones Through the Earth

4.3 Geodesic Curves

4.4 Problems with Variable End Points

4.5 How to Design a Thrilling Chute-the-Chute

4.6 Functionals Involving Several Unknown Functions

4.7 Fermat's Principle in Geometrical Optics

4.8 Hamilton's Principle of Stationary Action an Example on Small Vibrations

4.9 The McShane-Blankinship Curtain Rod Problem Functionals Involving Higher-Order Derivatives

4.10 Functionals Involving Several Independent Variables the Minimal Surface Problem

4.11 The Vibrating String

5. Applications of the Euler-Lagrange Multiplier Theorem to Problems with Global Pointwise Inequality Constraints

5.1 Slack Functions and Composite Curves

5.2 An Optimum Consumption Policy with Terminal Savings Constraint Without Extreme Hardship

5.3 A Problem in Production Planning with Inequality Constraints

6. Applications of the Euler-Lagrange Multiplier Theorem in Elementary Control Theory

6.1 Introduction

6.2 A Rocket Control Problem: Minimum Time

6.3 A Rocket Control Problem: Minimum Fuel

6.4 A More General Control Problem

6.5 A Simple Bang-Bang Problem

6.6 Some Remarks on the Maximum Principle and Dynamic Programming

7. The Variational Description of Sturm-Liouville Eigenvalues

7.1 Introduction to Sturm-Liouville Problems

7.2 The Relation Between the Lowest Eigenvalue and the Rayleigh Quotient

7.3 The Rayleigh-Ritz Method for the Lowest Eigenvalue

7.4 Higher Eigenvalues and the Rayleigh Quotient

7.5 The Courant Minimax Principle

7.6 Some Implications of the Courant Minimax Principle

7.7 Further Extensions of the Theory

7.8 Some General Remarks on the Ritz Method of Approximate Minimization

8. Some Remarks on the Use of the Second Variation in Extremum Problems

8.1 Higher-Order Variations

8.2 A Necessary Condition Involving the Second Variation at an Extremum

8.3 Sufficient Conditions for a Local Extremum

Appendix 1. The Cauchy and Schwarz Inequalities

Appendix 2. An Example on Normed Vector Spaces

Appendix 3. An Integral Inequality

Appendix 4. A Fundamental Lemma of the Calculus of Variations

Appendix 5. Du Bois-Reymond's Derivation of the Euler-Lagrange Equation

Appendix 6. A Useful Result from Calculus

Appendix 7. The Construction of a Certain Function

Appendix 8. The Fundamental Lemma for the Case of Several Independent Variables

Appendix 9. The Kinetic Energy for a Certain Model of an Elastic String

Appendix 10. The Variation of an Initial Value Problem with Respect to a Parameter

Subject Index Author Index

Library of Congress subject headings for this publication: Mathematical optimization, Calculus of variations