To the problem of a strong differentiability of integrals along different directions

1998 ◽  
Vol 5 (2) ◽  
pp. 157-176
Author(s):  
G. Lepsveridze
Keyword(s):  
Strong Differentiability

scholarly journals On a Characterisation of Differentiability of the Norm of a Normed Linear Space

1971 ◽  
Vol 12 (1) ◽  
pp. 106-114 ◽  
Author(s):  
J. R. Giles
Keyword(s):  
Banach Space ◽  
Linear Space ◽  
Normed Linear Space ◽  
Dual Norm ◽  
Uniform Rotundity ◽  
Strong Differentiability ◽  
Differentiability Properties

The purpose of this paper is to show that the various differentiability conditions for the norm of a normed linear space can be characterised by continuity conditions for a certain mapping from the space into its dual. Differentiability properties of the norm are often more easily handled using this characterisation and to demonstrate this we give somewhat more direct proofs of the reflexivity of a Banach space whose dual norm is strongly differentiable, and the duality of uniform rotundity and uniform strong differentiability of the norm for a Banach space.

Arcs of Parabolic Order Four

1964 ◽  
Vol 16 ◽  
pp. 321-338 ◽  
Author(s):  
N. D. Lane
Keyword(s):  
Affine Plane ◽  
Cyclic Order ◽  
Proved Theorem ◽  
Present Discussion ◽  
The Real ◽  
End Point ◽  
Real Affine ◽  
Order Four ◽  
Strong Differentiability

This paper is concerned with some of the properties of arcs in the real affine plane which are met by every parabola at not more than four points. Many of the properties of arcs of parabolic order four which we consider here are analogous to the corresponding properties of arcs of cyclic order three in the conformai plane which are described in (1). The paper (2), on parabolic differentiation, provides the background for the present discussion.In Section 2, general tangent, osculating, and superosculating parabolas are introduced. The concept of strong differentiability is introduced in Section 3; cf. Theorem 1. Section 4 deals with arcs of finite parabolic order, and it is proved (Theorem 2) that an end point p of an arc A of finite parabolic order is twice parabolically differentiable.

A fractional super-twisting control of electrically driven mechanical systems

2019 ◽  
Vol 42 (3) ◽  
pp. 485-492 ◽  
Author(s):  
Aldo Jonathan Muñoz-Vázquez ◽  
Juan Diego Sánchez-Torres ◽  
Vicente Parra-Vega ◽  
Anand Sánchez-Orta ◽  
Fernando Martínez-Reyes
Keyword(s):  
Structural Modification ◽  
Sliding Mode ◽  
Sliding Modes ◽  
Integer Order ◽  
Smooth Effects ◽  
Twisting Control ◽  
Strong Differentiability ◽  
Alternative Control Methods ◽  
Adequate Design ◽  
Super Twisting Control

The Super Twisting Control Algorithm (STA) constitutes a powerful and robust technique for control and observation problems. The structure of the STA allows inducing second-order sliding modes, such that the sliding variable and its derivative remain at zero after some finite time. However, the STA requires the strong differentiability of the sliding variable and the weak differentiability of disturbances. Thus, the sliding variable should become from an adequate design, ensuring its strong differentiability. Nonetheless, in the more general case of not necessarily integer-order differentiable disturbances, a typical case in electromechanical systems due to non-smooth effects, alternative control methods need to be considered. For that reason, this paper proposes a structural modification of the STA, allowing the integral of the discontinuous function to assume a fractional order to compensate not necessarily integer-order differentiable disturbances. An experimental assessment is conducted, and comparisons to other sliding mode based controllers are presented to demonstrate the reliability of the proposed method.

scholarly journals Strong differentiability with respect to product measures

1984 ◽  
Vol 78 (2) ◽  
pp. 173-178 ◽  
Author(s):  
N. Fava ◽  
O. Capri
Keyword(s):  
Product Measures ◽  
Strong Differentiability

scholarly journals Strong differentiability of the norm and rotundity of the dual

1978 ◽  
Vol 26 (3) ◽  
pp. 302-308 ◽  
Author(s):  
J. R. Giles
Keyword(s):  
Dual Space ◽  
Causal Relation ◽  
Linear Spaces ◽  
Normed Linear Spaces ◽  
Strong Differentiability

AbstractFor normed linear spaces two similar characterizations of strong differentiability of the norm and rotundity of the dual space are established, but it is shown that in general there is no causal relation between these two concepts.

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