**Student Mathematical Library**

Volume: 50;
2009;
252 pp;
Softcover

MSC: Primary 49;
Secondary 92

Print ISBN: 978-0-8218-4772-5

Product Code: STML/50

List Price: $48.00

Individual Price: $38.40

**Electronic ISBN: 978-1-4704-1635-5
Product Code: STML/50.E**

List Price: $48.00

Individual Price: $38.40

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#### Supplemental Materials

# A Primer on the Calculus of Variations and Optimal Control Theory

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*Mike Mesterton-Gibbons*

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.

#### Table of Contents

# Table of Contents

## A Primer on the Calculus of Variations and Optimal Control Theory

- Cover Cover11 free
- Title page iii4 free
- Contents vii8 free
- Foreword ix10 free
- Acknowledgments xiii14 free
- The Brachistochrone 116 free
- The fundamental problem. Extremals 722
- The insufficiency of extremality 1934
- Important first integrals 2944
- The du Bois-Reymond equation 3550
- The corner conditions 4156
- Legendre’s necessary condition 5166
- Jacobi’s necessary condition 5772
- Weak versus strong variations 6782
- Weierstrass’s necessary condition 7388
- The transversality conditions 8196
- Hilbert’s invariant integral 91106
- The fundamental sufficient condition 101116
- Jacobi’s condition revisited 111126
- Isoperimetrical problems 119134
- Optimal control problems 127142
- Necessary conditions for optimality 135150
- Time-optional control 149164
- A singular control problem 159174
- A biological control problem 163178
- Optimal control to a general target 167182
- Navigational control problems 183198
- State variable restrictions 195210
- Optimal harvesting 203218
- Afterword 219234
- Solutions or hints for selected exercises 221236
- Bibliography 245260
- Index 249264 free
- Back Cover Back Cover1274

#### Readership

Undergraduate and graduate students interested in the calculus of variations and optimal control.

#### Reviews

It is useful for libraries supporting applied mathematics programs or advanced course in [certain] disciplines.

-- CHOICE Magazine