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Lectures on the Calculus of Variations and Optimal Control Theory: Second Edition
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AMS Chelsea Publishing
1980; 337 pp; hardcover
Volume: 304
ISBN-10: 0-8218-2690-5
ISBN-13: 978-0-8218-2690-4
List Price: US$43 Member Price: US$38.70
Order Code: CHEL/304.H

This book is divided into two parts. The first addresses the simpler variational problems in parametric and nonparametric form. The second covers extensions to optimal control theory.

The author opens with the study of three classical problems whose solutions led to the theory of calculus of variations. They are the problem of geodesics, the brachistochrone, and the minimal surface of revolution. He gives a detailed discussion of the Hamilton-Jacobi theory, both in the parametric and nonparametric forms. This leads to the development of sufficiency theories describing properties of minimizing extremal arcs.

Next, the author addresses existence theorems. He first develops Hilbert's basic existence theorem for parametric problems and studies some of its consequences. Finally, he develops the theory of generalized curves and "automatic" existence theorems.

In the second part of the book, the author discusses optimal control problems. He notes that originally these problems were formulated as problems of Lagrange and Mayer in terms of differential constraints. In the control formulation, these constraints are expressed in a more convenient form in terms of control functions. After pointing out the new phenomenon that may arise, namely, the lack of controllability, the author develops the maximum principle and illustrates this principle by standard examples that show the switching phenomena that may occur. He extends the theory of geodesic coverings to optimal control problems. Finally, he extends the problem to generalized optimal control problems and obtains the corresponding existence theorems.

Graduate students and research mathematicians interested in calculus of variations and optimal control.

Reviews

"The appearance of this book is one of the most exciting events for friends of the Calculus of Variations since the publication of Carathéodory's classic in 1935 on the calculus of variations and partial differential equations of first order. The author ... gives here a very lively, greatly stimulating, and highly personalized account of the calculus of variations and optimal control theory ... In his many refreshing asides, the author not only puts ideas and techniques into their historic perspective, but also succeeds in making men, who for many of us are merely revered names, come alive through skillful selection of quotes and descriptions of their interaction with each other and the subject matter at hand ... A beautiful book ... that is bound to stimulate many mathematicians and students of mathematics."

-- MAA Monthly

"A considerable number of heretofore unpublished results developed by the author are found ... The book is an important contribution to the calculus of variations and optimal control theory. It is most appropriate that the theory of generalized curves should be presented ... by its founder. The book is well written with an unusual and lively style. It is filled with historical remarks and with comments which enlarge one's outlook on the role of mathematics and mathematicians in our society ... This book should be mastered by anyone who wishes to become an expert in this field."

-- Mathematical Reviews

Volume I. Lectures on the Calculus of Variations
• Generalities and typical problems
• The method of geodesic coverings
• Duality and local embedding
• Embedding in the large
• Hamiltonians in the large, convexity, inequalities and functional analysis
• Existence theory and its consequences
• Generalized curves and flows
• Appendix I: Some further basic notions of convexity and integration
• Appendix II: The variational significance and structure of generalized flows
Volume II. Optimal Control Theory
• The nature of control problems
• Naive optimal control theory
• The application of standard variational methods to optimal control
• Generalized optimal control
• References
• Index