scholarly journals A geometric application of a variation of Wirtinger inequality

1981 ◽  
Vol 106 (1) ◽  
pp. 42-47
Author(s):  
Zbyněk Nádeník
Keyword(s):  
Wirtinger Inequality ◽  
Geometric Application

Higher order Wirtinger inequalities

1980 ◽  
Vol 85 (1-2) ◽  
pp. 87-110 ◽  
Author(s):  
Kurt Kreith ◽  
Charles A. Swanson
Keyword(s):  
Boundary Conditions ◽  
Quadratic Forms ◽  
Differential Operators ◽  
Conjugate Point ◽  
Wirtinger Inequality ◽  
Even Order ◽  
Linear Differential ◽  
Natural Boundary Conditions ◽  
Derivatives Of ◽  
Admissible Functions

SynopsisWirtinger-type inequalities of order n are inequalities between quadratic forms involving derivatives of order k ≦ n of admissible functions in an interval (a, b). Several methods for establishing these inequalities are investigated, leading to improvements of classical results as well as systematic generation of new ones. A Wirtinger inequality for Hamiltonian systems is obtained in which standard regularity hypotheses are weakened and singular intervals are permitted, and this is employed to generalize standard inequalities for linear differential operators of even order. In particular second order inequalities of Beesack's type are developed, in which the admissible functions satisfy only the null boundary conditions at the endpoints of [a, b] and b does not exceed the first systems conjugate point (a) of a. Another approach is presented involving the standard minimization theory of quadratic forms and the theory of “natural boundary conditions”. Finally, inequalities of order n + k are described in terms of (n, n)-disconjugacy of associated 2nth order differential operators.

scholarly journals Stabilization of Discrete-Time Delayed Systems in Presence of Actuator Saturation Based on Wirtinger Inequality

2019 ◽  
Vol 2019 ◽  
pp. 1-14
Author(s):  
Vipin Chandra Pal ◽  
Richa Negi ◽  
Quanxin Zhu
Keyword(s):  
Discrete Time ◽  
Actuator Saturation ◽  
Convex Combination ◽  
Optimization Procedure ◽  
Linear Matrix ◽  
Delayed Systems ◽  
Delay Bound ◽  
Wirtinger Inequality ◽  
Time Control ◽  
Time Problem

This paper examines the stability analysis of discrete-time control systems particularly during the event of actuator saturation and time-varying state delay. With the help of Wirtinger inequality along with Lyapunov-Krasovskii functional gain of state feedback controller is determined for stabilization of above system. The saturation nonlinearity is represented in the terms of convex hull. A new linear matrix inequality (LMI) criterion is settled with reciprocally convex combination based inequality which is dependent on delay. The proposed criterion is less conservative in concern to increase the delay bound and a controller is also simulated for real time problem of missile control system in this paper. It is also attained that projected stability criterion is less conservative compared to other outcomes. Furthermore, an optimization procedure together with LMI constraints has been proposed to maximize the attraction of domain.

Simultaneous approximation of Birkhoff interpolation and the associated sharp inequalities

2020 ◽  
Vol 18 (04) ◽  
pp. 2050021
Author(s):  
Zehong Liu ◽  
Wanting Lu ◽  
Guiqiao Xu
Keyword(s):  
Integral Operator ◽  
Simultaneous Approximation ◽  
Approximation Error ◽  
Interpolation Formula ◽  
Maximum Eigenvalue ◽  
Birkhoff Interpolation ◽  
Wirtinger Inequality ◽  
Interpolation Matrix ◽  
Schmidt Operator ◽  
Picone Inequality

This paper gives a kind of sharp simultaneous approximation error estimation of Birkhoff interpolation [Formula: see text], [Formula: see text] where [Formula: see text] and [Formula: see text] is the Birkhoff interpolation based on [Formula: see text] pairs of numbers [Formula: see text] with its P[Formula: see text]lya interpolation matrix to be regular. First, based on the integral remainder formula of Birkhoff interpolation, we refer the computation of [Formula: see text] to the norm of an integral operator. Second, we refer the values of [Formula: see text] and [Formula: see text] to two explicit integral expressions and the value of [Formula: see text] to the computation of the maximum eigenvalue of a Hilbert–Schmidt operator. At the same time, we give the corresponding sharp Wirtinger inequality [Formula: see text] and sharp Picone inequality [Formula: see text].

Robust mixed H2 and passive switching control for uncertain discrete switched systems with time delay

2019 ◽  
Vol 37 (2) ◽  
pp. 422-440 ◽  
Author(s):  
Chang-Hua Lien ◽  
Ker-Wei Yu ◽  
Hao-Chin Chang
Keyword(s):  
Time Delay ◽  
Switched Systems ◽  
Switching Control ◽  
Delay System ◽  
Lyapunov Theory ◽  
Linear Matrix ◽  
Time Delay System ◽  
Wirtinger Inequality ◽  
Discrete Time Delay ◽  
Discrete Switched Systems

Abstract In this paper, the problem of mixed ${H}_2$ and passive switching control of uncertain discrete time-delay switched systems is investigated via a switching signal selection. Lyapunov theory with Wirtinger inequality is applied to guarantee the mixed performance for discrete switched time-delay system. The used Linear Matrix Inequality variables are less than our past proposed results. Finally, the improvement of the developed results is illustrated via a numerical example.

scholarly journals On the matrix Monge–Kantorovich problem

2019 ◽  
Vol 31 (4) ◽  
pp. 574-600 ◽  
Author(s):  
YONGXIN CHEN ◽  
WILFRID GANGBO ◽  
TRYPHON T. GEORGIOU ◽  
ALLEN TANNENBAUM
Keyword(s):  
Euclidean Distance ◽  
Strong Duality ◽  
Wasserstein Distance ◽  
Positive Definite ◽  
Type Result ◽  
Density Matrices ◽  
Wirtinger Inequality ◽  
The Matrix ◽  
The Cost ◽  
Kantorovich Problem

The classical Monge–Kantorovich (MK) problem as originally posed is concerned with how best to move a pile of soil or rubble to an excavation or fill with the least amount of work relative to some cost function. When the cost is given by the square of the Euclidean distance, one can define a metric on densities called the Wasserstein distance. In this note, we formulate a natural matrix counterpart of the MK problem for positive-definite density matrices. We prove a number of results about this metric including showing that it can be formulated as a convex optimisation problem, strong duality, an analogue of the Poincaré–Wirtinger inequality and a Lax–Hopf–Oleinik–type result.

New delay-range-dependent stability condition for fuzzy Hopfield neural networks via Wirtinger inequality

2020 ◽  
Vol 38 (5) ◽  
pp. 6099-6109
Author(s):  
Rupak Datta ◽  
Rajeeb Dey ◽  
Ramasamy Saravanakumar ◽  
Baby Bhattacharya ◽  
Tsung-Chih Lin
Keyword(s):  
Neural Networks ◽  
Stability Condition ◽  
Hopfield Neural Networks ◽  
Wirtinger Inequality
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