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Marston Morse applied calculus of variations in what is now called Morse theory. [6] Lev Pontryagin, Ralph Rockafellar and F. H. Clarke developed new mathematical tools for the calculus of variations in optimal control theory. [6] The dynamic programming of Richard Bellman is an alternative to the calculus of variations. [7] [8] [9] [c]
In mathematics, specifically in the calculus of variations, a variation δf of a function f can be concentrated on an arbitrarily small interval, but not a single point. Accordingly, the necessary condition of extremum ( functional derivative equal zero) appears in a weak formulation (variational form) integrated with an arbitrary function δf .
In mathematics, the direct method in the calculus of variations is a general method for constructing a proof of the existence of a minimizer for a given functional, [1] introduced by Stanisław Zaremba and David Hilbert around 1900. The method relies on methods of functional analysis and topology. As well as being used to prove the existence of ...
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In the calculus of variations, functionals are usually expressed in terms of an integral of functions, their arguments, and their derivatives. In an integrand L of a functional, if a function f is varied by adding to it another function δf that is arbitrarily small, and the resulting integrand is expanded in powers of δf , the coefficient of ...
In the field of mathematical analysis for the calculus of variations, Γ-convergence (Gamma-convergence) is a notion of convergence for functionals. It was introduced by Ennio De Giorgi . Definition
This was a major unsolved problem in the Calculus of Variations, until Šverák gave an counterexample in 1993 for the case and . [11] The case d = 2 {\displaystyle d=2} or m = 2 {\displaystyle m=2} is still an open problem, known as Morrey's conjecture.
Malliavin calculus is also called the stochastic calculus of variations. P. Malliavin first initiated the calculus on infinite dimensional space. Then, the significant contributors such as S. Kusuoka, D. Stroock, J-M. Bismut, Shinzo Watanabe, I. Shigekawa, and so on finally completed the foundations.