User:Julius/Sandbox

Subject Matter

 * Calculus of Variations

Contents

 * I. Introduction


 * II. The First Variation


 * III. Some Generalizations


 * IV. Isoperimetric Problems


 * V. Applications to Eigenvalue Problems


 * VI. Holonomic and Nonholonomic Constraints


 * VII. Problems with Variable Endpoints


 * VIII. The Hamiltonian Formulation


 * IX. Noether's Theorem


 * X. The Second Variation


 * A. Analysis and Differential Equations


 * B. Function Spaces


 * References


 * Index

Subject Matter

 * Calculus of Variations

Contents
Preface

Acknowledgments


 * I. Calculus of Variations


 * 1. Historical Notes on the Calculus of Variations


 * 2. Introduction and Preliminaries


 * 3. The Simplest Problem in the Calculus of Variations


 * 4. Necessary Conditions for Local Minima


 * 5. Sufficient Conditions for the Simplest Problem


 * 6. Summary for the Simplest Problem


 * 7. Extensions and Generalizations


 * 8. Applications


 * II. Optimal Control


 * 9. Optimal Control Problems


 * 10. Simplest Problem in Optimal Control


 * 11. Extensions of the Maximum Principle


 * 12. Linear Control Systems

Bibliography

Index

Subject Matter

 * Calculus of Variations

Contents
Preface

Acknowledgments


 * I. Calculus of Variations


 * 1. Historical Notes on the Calculus of Variations


 * 2. Introduction and Preliminaries


 * 3. The Simplest Problem in the Calculus of Variations


 * 4. Necessary Conditions for Local Minima


 * 5. Sufficient Conditions for the Simplest Problem


 * 6. Summary for the Simplest Problem


 * 7. Extensions and Generalizations


 * 8. Applications


 * II. Optimal Control


 * 9. Optimal Control Problems


 * 10. Simplest Problem in Optimal Control


 * 11. Extensions of the Maximum Principle


 * 12. Linear Control Systems

Bibliography

Index

Subject Matter

 * Calculus of Variations

Contents
Symbols and Notation

PART 1. CALCULUS OF VARITAIONS

1. Introduction

2. Problem Statement and Necessary Conditions for an Extremum

3. Integration of the Euler - Lagrange Equation

4. An Inverse Problem

5. The Weierstrass Necessary Condition

6. Jacobi's Necessary Condition

7. Corner Conditions

8. Concluding Remarks

PART II. OPTIMAL CONTROL

9. Introduction

10. Problem Statement and Optimality

11. Regular Optimal Trajectories

12. Examples of Extremal Control

13. Some Generalizations

14. Special Systems

15. Sufficient Conditions

16. Feedback Control

17. Optimization with Vector - Valued Cost

REFERENCES

BIBLIOGRAPHY

INDEX

Subject Matter

 * Calculus of Variations

Contents
Part I: Measure Theory and $ L^p $ Spaces

1. Measures

2. $ L^p $ Spaces

Part II: The Direct Method and Lower Semicontinuity

3. The Direct Method and Lower Semicontinuity

4. Convex Analysis

Part III: Functional Defined on $ L^p $

5. Integrands $ f = f \left ( { z } \right ) $

6. Integrands $ f = f \left ( { x, z } \right ) $

7. Integrands $ f = f \left ( { x, u, z } \right ) $

8. Young Measures

Part IV: Appendix

A. Functional Analysis and Set Theory

B. Notes and Open Problems

Notations and List of Symbols

Acknowledgments

References

Index