A Primer on the Calculus of Variations and Optimal Control Theory (Student Mathematical Library) (Student Mathematical Library, 50)

The calculus of variations is used to find functions that optimize quantities expressed in terms of integrals. Optimal control theory seeks to find functions that minimize cost integrals for systems described by differential equations. This book is an introduction to both the classical theory of the calculus of variations and the more modern developments of optimal control theory from the perspective of an applied mathematician. It focuses on understanding concepts and how to apply them. The range of potential applications is broad: the calculus of variations and optimal control theory have been widely used in numerous ways in biology, criminology, economics, engineering, finance, management science, and physics. Applications described in this book include cancer chemotherapy, navigational control, and renewable resource harvesting. The prerequisites for the book are modest: the standard calculus sequence, a first course on ordinary differential equations, and some facility with the use of mathematical software. It is suitable for an undergraduate or beginning graduate course, or for self study. It provides excellent preparation for more advanced books and courses on the calculus of variations and optimal control theory. The Calculus Of Variations Is Used To Find Functions That Optimize Quantities Expressed In Terms Of Integrals. Optimal Control Theory Seeks To Find Functions That Minimize Cost Integrals For Systems Described By Differential Equations. This Book Is An Introduction To Both The Classical Theory Of The Calculus Of Variations And The More Modern Developments Of Optimal Control Theory From The Perspective Of An Applied Mathematician. It Focuses On Understanding Concepts And How To Apply Them. The Range Of Potential Applications Is Broad: The Calculus Of Variations And Optimal Control Theory Have Been Widely Used In Numerous Ways In Biology, Criminology, Economics, Engineering, Finance, Management Science, And Physics. Applications Described In This Book Include Cancer Chemotherapy, Navigational Control, And Renewable Resource Harvesting.--page 4 Of Cover. Lecture 1. The Brachistochrone Lecture 2. The Fundamental Problem. Extremals Lecture 3. The Insufficiency Of Extremality Lecture 4. Important First Integrals Lecture 5. The Du Bois-reymond Equation Lecture 6. The Corner Conditions Lecture 7. Legendre's Necessary Condition Lecture 8. Jacobi's Necessary Condition Lecture 9. Weak Versus Strong Variations Lecture 10. Weierstrass's Necessary Condition Lecture 11. The Transversality Conditions Lecture 12. Hilbert's Invariant Integral Lecture 13. The Fundamental Sufficient Condition Lecture 14. Jacobi's Condition Revisited Lecture 15. Isoperimetrical Problems Lecture 16. Optimal Control Problems Lecture 17. Necessary Conditions For Optimality Lecture 18. Time-optimal Control Lecture 19. A Singular Control Problem Lecture 20. A Biological Control Problem Lecture 21. Optimal Control To A General Target Lecture 22. Navigational Control Problems Lecture 23. State Variable Restrictions Lecture 24. Optimal Harvesting Afterword Solutions Or Hints For Selected Exercises Mike Mesterton-gibbons. Includes Bibliographical References (p. 245-248) And Index. Cover 1 Title page 4 Contents 8 Foreword 10 Acknowledgments 14 The Brachistochrone 16 The fundamental problem. Extremals 22 The insufficiency of extremality 34 Important first integrals 44 The du Bois-Reymond equation 50 The corner conditions 56 Legendre’s necessary condition 66 Jacobi’s necessary condition 72 Weak versus strong variations 82 Weierstrass’s necessary condition 88 The transversality conditions 96 Hilbert’s invariant integral 106 The fundamental sufficient condition 116 Jacobi’s condition revisited 126 Isoperimetrical problems 134 Optimal control problems 142 Necessary conditions for optimality 150 Time-optional control 164 A singular control problem 174 A biological control problem 178 Optimal control to a general target 182 Navigational control problems 198 State variable restrictions 210 Optimal harvesting 218 Afterword 234 Solutions or hints for selected exercises 236 Bibliography 260 Index 264 Back Cover 274 Offers an introduction to both the classical theory of the calculus of variations and the more modern developments of optimal control theory from the perspective of an applied mathematician. This book focuses on understanding concepts and how to apply them. It describes applications such as cancer chemotherapy, and navigational control.