Neuware -Although the calculus of variations has ancient origins in questions of Ar istotle and Zenodoros, its mathematical principles first emerged in the post calculus investigations of Newton, the Bernoullis, Euler, and Lagrange. Its results now supply fundamental tools of exploration to both mathematicians and those in the applied sciences. (Indeed, the macroscopic statements ob tained through variational principles may provide the only valid mathemati cal formulations of many physical laws. ) Because of its classical origins, variational calculus retains the spirit of natural philosophy common to most mathematical investigations prior to this century. The original applications, including the Bernoulli problem of finding the brachistochrone, require opti mizing (maximizing or minimizing) the mass, force, time, or energy of some physical system under various constraints. The solutions to these problems satisfy related differential equations discovered by Euler and Lagrange, and the variational principles of mechanics (especially that of Hamilton from the last century) show the importance of also considering solutions that just provide stationary behavior for some measure of performance of the system. However, many recent applications do involve optimization, in particular, those concerned with problems in optimal control. Optimal control is the rapidly expanding field developed during the last half-century to analyze optimal behavior of a constrained process that evolves in time according to prescribed laws. Its applications now embrace a variety of new disciplines, including economics and production planning.Springer Verlag GmbH, Tiergartenstr. 17, 69121 Heidelberg 482 pp. Englisch.

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Titel: Variational Calculus and Optimal Control
Verlag: Springer New York, Springer US Dez 1995
Ver�ffentlichungsdatum: 1995
Sprache: Englisch
ISBN-10: 0387945113
ISBN-13: 9780387945118
Bindung: Buch
Ausgabe: 2. Auflage
Zustand: Neu

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An introduction to the variational methods used to formulate and solve mathematical and physical problems, allowing the reader an insight into the systematic use of elementary (partial) convexity of differentiable functions in Euclidian space. By helping students directly characterize the solutions for many minimization problems, the text serves as a prelude to the field theory for sufficiency, laying as it does the groundwork for further explorations in mathematics, physics, mechanical and electrical engineering, as well as computer science.

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This book supplies a broad-based introduction to variational methods for formulating and solving problems in mathematics and the applied sciences. It refines and extends the author's earlier text on variational calculus and a supplement on optimal control. It is the only current introductory text that uses elementary partial convexity of differentiable functions to characterize directly the solutions of some minimization problems before exploring necessary conditions for optimality or field theory methods of sufficiency. Through effective notation, it combines rudiments of analysis in (normed) linear spaces with simpler aspects of convexity to offer a multilevel strategy for handling such problems. It also employs convexity considerations to broaden the discussion of Hamilton's principle in mechanics and to introduce Pontjragin's principle in optimal control. It is mathematically self-contained but it uses applications from many disciplines to provide a wealth of examples and exercises. The book is accessible to upper-level undergraduates and should help its user understand theories of increasing importance in a society that values optimal performance.

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