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 Brachistochrone * The fundamental problem. Extremals * The insufficiency of extremality * Important first integrals * The du Bois-Reymond equation * The corner conditions * Legendre’s necessary condition * Jacobi’s necessary condition * Weak versus strong variations * Weierstrass’s necessary condition * The transversality conditions * Hilbert’s invariant integral * The fundamental sufficient condition * Jacobi’s condition revisited * Isoperimetrical problems * Optimal control problems * Necessary conditions for optimality * Time-optional control * A singular control problem * A biological control problem * Optimal control to a general target * Navigational control problems * State variable restrictions * Optimal harvesting * Afterword * Solutions or hints for selected exercises * Bibliography * Index