scholarly journals The classical theory of calculus of variations for generalized functions

2017 ◽  
Vol 8 (1) ◽  
pp. 779-808 ◽  
Author(s):  
Alexander Lecke ◽  
Lorenzo Luperi Baglini ◽  
Paolo Giordano
Keyword(s):  
Calculus Of Variations ◽  
Classical Theory ◽  
Sufficient Conditions ◽  
Generalized Functions ◽  
Conjugate Points ◽  
Lagrange Equations ◽  
Smooth Functions ◽  
Nonlinear Properties ◽  
Schwartz Distributions ◽  
Necessary And Sufficient

Abstract We present an extension of the classical theory of calculus of variations to generalized functions. The framework is the category of generalized smooth functions, which includes Schwartz distributions, while sharing many nonlinear properties with ordinary smooth functions. We prove full connections between extremals and Euler–Lagrange equations, classical necessary and sufficient conditions to have a minimizer, the necessary Legendre condition, Jacobi’s theorem on conjugate points and Noether’s theorem. We close with an application to low regularity Riemannian geometry.

scholarly journals Variational Problems with Partial Fractional Derivative: Optimal Conditions and Noether’s Theorem

2018 ◽  
Vol 2018 ◽  
pp. 1-14
Author(s):  
Jun Jiang ◽  
Yuqiang Feng ◽  
Shougui Li
Keyword(s):  
Fractional Derivative ◽  
Sufficient Conditions ◽  
Variational Problems ◽  
Necessary And Sufficient Conditions ◽  
Optimal Conditions ◽  
Lagrange Equations ◽  
Noether’S Theorem ◽  
Noether's Theorem ◽  
Necessary And Sufficient

In this paper, the necessary and sufficient conditions of optimality for variational problems with Caputo partial fractional derivative are established. Fractional Euler-Lagrange equations are obtained. The Legendre condition and Noether’s theorem are also presented.

scholarly journals Alternative Lagrangians obtained by scalar deformations

2020 ◽  
Vol 17 (04) ◽  
pp. 2050050
Author(s):  
Oana A. Constantinescu ◽  
Ebtsam H. Taha
Keyword(s):  
Force Field ◽  
Differentiable Function ◽  
Mechanical Systems ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Initial System ◽  
Lagrange Equations ◽  
Necessary And Sufficient ◽  
Equations Of Motions

We study mechanical systems that can be recast into the form of a system of genuine Euler–Lagrange equations. The equations of motions of such systems are initially equivalent to the system of Lagrange equations of some Lagrangian [Formula: see text], including a covariant force field. We find necessary and sufficient conditions for the existence of a differentiable function [Formula: see text] such that the initial system is equivalent to the system of Euler–Lagrange equations of the deformed Lagrangian [Formula: see text].

scholarly journals Representation of functions as the Post-Widder inversion operator of generalized functions

1984 ◽  
Vol 7 (2) ◽  
pp. 371-396 ◽  
Author(s):  
R. P. Manandhar ◽  
L. Debnath
Keyword(s):  
Representation Theory ◽  
Function Space ◽  
Fundamental Theorem ◽  
Sufficient Conditions ◽  
Generalized Functions ◽  
Representation Theorems ◽  
The Real ◽  
Inversion Operator ◽  
Inversion Theory ◽  
Necessary And Sufficient

A study is made of the Post-Widder inversion operator to a class of generalized functions in the sense of distributional convergence. Necessary and sufficient conditions are proved for a given function to have the representation as therth operate of the Post-Widder inversion operator of generalized functions. Some representation theorems are also proved. Certain results concerning the testing function space and its dual are established. A fundamental theorem regarding the existence of the real inversion operator (1.6) withr=0is proved in section4. A classical inversion theory for the Post-Widder inversion operator with a few other theorems which are fundamental to the representation theory is also developed in this paper.

scholarly journals Automatic choice of parameters at approximation of functions by rational splines

1987 ◽  
pp. 52
Author(s):  
A.D. Malysheva
Keyword(s):  
Sufficient Conditions ◽  
Order Of Approximation ◽  
Approximation Of Functions ◽  
Smooth Functions ◽  
Higher Derivatives ◽  
Necessary And Sufficient ◽  
Automatic Choice ◽  
Derivatives Of ◽  
Best Parameters ◽  
Approximation Of A Function

We obtain necessary and sufficient conditions put on the parameters of rational splines that provide given order of approximation of smooth functions. We point out the formulas of asymptotically the best parameters of rational splines that, while providing the best order of approximation of a function by rational splines, do not contain information about the values of higher derivatives of a function.

scholarly journals Hypoelliptic differential operators with generalized constant coefficients

1998 ◽  
Vol 41 (1) ◽  
pp. 47-60 ◽  
Author(s):  
M. Nedeljkov ◽  
S. Pilipović
Keyword(s):  
Small Parameter ◽  
Differential Operators ◽  
Sufficient Conditions ◽  
Generalized Functions ◽  
Necessary And Sufficient Conditions ◽  
Order Estimate ◽  
Colombeau Generalized Functions ◽  
Constant Coefficients ◽  
Necessary And Sufficient

The space of Colombeau generalized functions is used as a frame for the study of hypoellipticity of a family of differential operators whose coefficients depend on a small parameter ε.There are given necessary and sufficient conditions for the hypoellipticity of a family of differential operators with constant coefficients which depend on ε and behave like powers of ε as ε→0. The solutions of such family of equations should also satisfy the power order estimate with respect to ε.

The Brachistochrone With a Movable End-Point and the Nonsimultaneous Variations

2009 ◽  
Vol 5 (1) ◽  
Author(s):  
D. Radomirovic ◽  
Dj. Djukic ◽  
L. Cveticanin
Keyword(s):  
Exact Solution ◽  
Calculus Of Variations ◽  
Parametric Method ◽  
Sufficient Conditions ◽  
Integral Type ◽  
First Order ◽  
End Point ◽  
Order Variation ◽  
Ritz’S Method ◽  
Necessary And Sufficient

In this paper, the necessary and sufficient conditions for minima plane path with a movable end-point are developed. Using the calculus of variations the considered conditions are based on the zero first-order nonsimultaneous variation and on the positive second-order variation in the functional of integral type corresponding to mechanical systems. The applied procedure is the coordinate parametric method. The obtained solutions are tested on a brachistochrone with one end-point constrained to lie on a circle. The exact solution is compared with the approximate one obtained with Ritz’s method.

Sufficiency and the Jacobi Condition in the Calculus of Variations

1986 ◽  
Vol 38 (5) ◽  
pp. 1199-1209 ◽  
Author(s):  
Frank H. Clarke ◽  
Vera Zeidan
Keyword(s):  
Calculus Of Variations ◽  
Basic Problem ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Jacobi Condition ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Differentiable Mappings

Besides stating the problem and the results, we shall give in this section a brief overview of the classical necessary and sufficient conditions in the calculus of variations, in order to clearly situate the contribution of this article.1.1 The problem. We are given an interval [a, b], two points xa, xb in Rn, and a function L (the Lagrangian) mapping [a, b] × Rn × Rn to R. The basic problem in the calculus of variations, labeled (P), is that of minimizing the functionalover some class X of functions x and subject to the constraints x(a) = xa, x(b) = xb. Let us take for now the class X of functions to be the continuously differentiable mappings from [a, b] to Rn; we call such functions smooth arcs.

scholarly journals Disturbance Decoupling Problem: Logic-Dynamic Approach-Based Solution

Symmetry ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 555 ◽  
Author(s):  
Alexey Zhirabok
Keyword(s):  
Control Systems ◽  
Sufficient Conditions ◽  
Algebraic Approach ◽  
Dynamic Measurement ◽  
Dynamic Approach ◽  
Smooth Functions ◽  
System Description ◽  
Disturbance Decoupling ◽  
Measurement Feedback ◽  
Necessary And Sufficient

This paper considers the disturbance decoupling problem by the dynamic measurement feedback for discrete-time nonlinear control systems. To solve this problem, the algebraic approach, called the logic-dynamic approach, is used. Such an approach assumes that the system description may contain non-smooth functions. Necessary and sufficient conditions are obtained in terms of matrices similar to controlled and ( h , f ) -invariant functions. Furthermore, procedures are developed to determine the corresponding matrices and the dynamic measurement feedback.

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