Titlebook: Variational Calculus and Optimal Control; Optimization with El John L. Troutman Textbook 1996Latest edition Springer Science+Business Media

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Minimization of Convex Functionsnear space ., such that a convex function is automatically minimized by a . ∈ . at which its Gâteaux variations vanish.. Moreover, in the presence of strict convexity, there can be at most one such .. A large and useful class of functions is shown to be convex. In particular, in §3.2, the role of [s

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The Euler-Lagrange Equations. However, it was not until the work of Euler (c. 1742) and Lagrange (1755) that the systematic theory now known as the calculus of variations emerged. Initially, it was restricted to finding conditions which were . in order that an integral function.should have a (local) extremum on a set. ⊆ {. ∈ .

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Variational Principles in Mechanicstions of Euler-Lagrange for the minimizing function made it natural for mathematicians of the eighteenth century to ask for an integral quantity whose minimization would result in Newton’s equations of motion. With such a quantity, a new principle through which the universe acts would be obtained. T

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Sufficient Conditions for a Minimumn . on set such as . = {. ∈ .([.]).: .(.) = .(.) = .}, since they are only conditions for the stationary of .. However, in the presence of [strong] convexity of .(.,.) these conditions do characterize [unique] minimization. [Cf.§3.2, Problem 3.33 et seq.] Not all such functions are convex, but we ha

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Control Problems and Sufficiency Considerationstus which was efficiently self-correcting, relative to some targeted objective. Such efficiency is clearly desirable in, say, the tracking of an aircraft near a busy airport or in the consumption of its fuel, and other economically desirable objectives suggest themselves. The underlying mathematical

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Necessary Conditions for Optimalityerval and its defining functions are suitably convex, then the methods of variational calculus can be adapted to suggest sufficient conditions for an optimal control. In particular, the minimum principle of §10.3 and §10.4 can guarantee optimality of a solution to the problem. In §11.1 we will disco

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Textbook 1996Latest editionst­ 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 mathe

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