scholarly journals A spherical rearrangement proof of the stability of a Riesz-type inequality and an application to an isoperimetric type problem

2021 ◽  
Author(s):  
Giacomo Ascione
Keyword(s):  
Isoperimetric Inequality ◽  
Type Inequality ◽  
Stability Result ◽  
Global Minimizer ◽  
Mass Transportation ◽  
Volume Constraint ◽  
Type Problem ◽  
Fixed Volume ◽  
The Stability ◽  
Perimeter Penalization

We prove the stability of the ball as global minimizer of an attractive shape functional under volume constraint, by means of mass transportation arguments. The stability exponent is $1/2$ and it is sharp. Moreover, we use such stability result together with the quantitative (possibly fractional) isoperimetric inequality to prove that the ball is a global minimizer of a shape functional involving both an attractive and a repulsive term with a sufficiently large fixed volume and with a suitable (possibly fractional) perimeter penalization.

Global stability result for parabolic Cauchy problems

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Mourad Choulli ◽  
Masahiro Yamamoto
Keyword(s):  
New York ◽  
Partial Differential Equations ◽  
Global Stability ◽  
Classical Problem ◽  
Stability Result ◽  
Cauchy Problems ◽  
Smooth Solutions ◽  
Linear Partial Differential Equations ◽  
Ill Posed ◽  
The Stability

AbstractUniqueness of parabolic Cauchy problems is nowadays a classical problem and since Hadamard [Lectures on Cauchy’s Problem in Linear Partial Differential Equations, Dover, New York, 1953], these kind of problems are known to be ill-posed and even severely ill-posed. Until now, there are only few partial results concerning the quantification of the stability of parabolic Cauchy problems. We bring in the present work an answer to this issue for smooth solutions under the minimal condition that the domain is Lipschitz.

L 1 and L ∞ stability of transition densities of perturbed diffusions

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Ilya Bitter ◽  
Valentin Konakov
Keyword(s):  
Linear Growth ◽  
Stability Result ◽  
Regularity Conditions ◽  
Initial Condition ◽  
Weak Regularity ◽  
Transition Densities ◽  
The Stability ◽  
Drift Terms

Abstract In this paper, we derive a stability result for L 1 {L_{1}} and L ∞ {L_{\infty}} perturbations of diffusions under weak regularity conditions on the coefficients. In particular, the drift terms we consider can be unbounded with at most linear growth, and the estimates reflect the transport of the initial condition by the unbounded drift through the corresponding flow. Our approach is based on the study of the distance in L 1 {L_{1}} - L 1 {L_{1}} metric between the transition densities of a given diffusion and the perturbed one using the McKean–Singer parametrix expansion. In the second part, we generalize the well-known result on the stability of diffusions with bounded coefficients to the case of at most linearly growing drift.

Output Feedback of Uncertain Multi-Input Systems

2020 ◽  
pp. 169-198
Author(s):  
Yang Zhu ◽  
Miroslav Krstic
Keyword(s):  
Output Feedback ◽  
Stability Result ◽  
Initial State ◽  
Linear Controller ◽  
Single Input ◽  
The Stability ◽  
Input Delays ◽  
Local Result ◽  
Practical Sense ◽  
Initial Delay

This chapter evaluates output feedback of uncertain multi-input systems. Similar to the case of single-input delay, the result of multi-input delays obtained in the chapter is not global, as it does not believe the problem where the actuator state is not measurable and the delay value is unknown at the same time is solvable globally, since the problem is not linearly parameterized. In a practical sense, the stability result proven in the chapter is not a highly satisfactory result since it is local both in the initial state and in the initial parameter error. This means that the initial delay estimate needs to be sufficiently close to the true delay. Under such an assumption, one might as well use a linear controller and rely on robustness of the feedback law to small errors in the assumed delay value. Nevertheless, the chapter presents the local result here as it highlights quite clearly why a global result is not obtainable when both the delay value and the delay state are unavailable.

POSITIVE CENTRE SETS OF CONVEX CURVES AND A BONNESEN TYPE INEQUALITY

2018 ◽  
Vol 99 (2) ◽  
pp. 293-301
Author(s):  
YUNLONG YANG
Keyword(s):  
Isoperimetric Inequality ◽  
Type Inequality ◽  
Convex Curves

We consider the positive centre sets of regular $n$-gons, rectangles and half discs, and conjecture a Bonnesen type inequality concerning positive centre sets which is stronger than the classical isoperimetric inequality.

scholarly journals Stability of Riemannian manifolds with Killing spinors

2017 ◽  
Vol 28 (01) ◽  
pp. 1750005 ◽  
Author(s):  
Changliang Wang
Keyword(s):  
Riemannian Manifolds ◽  
Einstein Manifold ◽  
Stability Result ◽  
Warped Products ◽  
Einstein Manifolds ◽  
Killing Spinors ◽  
Parallel Spinors ◽  
Unstable Manifolds ◽  
The Stability ◽  
Complete Riemannian Manifolds

Riemannian manifolds with nonzero Killing spinors are Einstein manifolds. Kröncke proved that all complete Riemannian manifolds with imaginary Killing spinors are (linearly) strictly stable in [Stable and unstable Einstein warped products, preprint (2015), arXiv:1507.01782v1 ]. In this paper, we obtain a new proof for this stability result by using a Bochner-type formula in [X. Dai, X. Wang and G. Wei, On the stability of Riemannian manifold with parallel spinors, Invent. Math. 161(1) (2005) 151–176; M. Wang, Preserving parallel spinors under metric deformations, Indiana Univ. Math. J. 40 (1991) 815–844]. Moreover, existence of real Killing spinors is closely related to the Sasaki–Einstein structure. A regular Sasaki–Einstein manifold is essentially the total space of a certain principal [Formula: see text]-bundle over a Kähler–Einstein manifold. We prove that if the base space is a product of two Kähler–Einstein manifolds then the regular Sasaki–Einstein manifold is unstable. This provides us many new examples of unstable manifolds with real Killing spinors.

scholarly journals STABILITY OF CRYSTALLINE EVOLUTIONS

2005 ◽  
Vol 15 (06) ◽  
pp. 921-937 ◽  
Author(s):  
MATTEO NOVAGA ◽  
EMANUELE PAOLINI
Keyword(s):  
Integral Curve ◽  
Jacobian Matrix ◽  
Convex Polyhedra ◽  
Wulff Shape ◽  
Stability Properties ◽  
Fixed Volume ◽  
The Stability

In this paper we analyze the stability properties of the Wulff-shape in the crystalline flow. It is well known that the Wulff-shape evolves self-similarly, and eventually shrinks to a point. We consider the flow restricted to the set of convex polyhedra, we show that the crystalline evolutions may be viewed, after a proper rescaling, as an integral curve in the space of polyhedra with fixed volume, and we compute the Jacobian matrix of this field. If the eigenvalues of such a matrix have real part different from zero, we can determine if the Wulff-shape is stable or unstable, i.e. if all the evolutions starting close enough to the Wulff-shape converge or not, after rescaling, to the Wulff-shape itself.

The stability of an additive functional equation in menger probabilistic φ-normed spaces

2011 ◽  
Vol 61 (5) ◽  
Author(s):  
D. Miheţ ◽  
R. Saadati ◽  
S. Vaezpour
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
Fixed Point Theory ◽  
Stability Result ◽  
Point Theory ◽  
Additive Functional ◽  
Normed Spaces ◽  
Additive Functional Equation ◽  
Probabilistic Normed Spaces ◽  
The Stability

AbstractWe establish a stability result concerning the functional equation: $\sum\limits_{i = 1}^m {f\left( {mx_i + \sum\limits_{j = 1,j \ne i}^m {x_j } } \right) + f\left( {\sum\limits_{i = 1}^m {x_i } } \right) = 2f\left( {\sum\limits_{i = 1}^m {mx_i } } \right)} $ in a large class of complete probabilistic normed spaces, via fixed point theory.

Study on Information Science with Stability Analysis for a Class of Interval Linear Networked Control System

2013 ◽  
Vol 703 ◽  
pp. 231-235
Author(s):  
Yang Gao ◽  
Wei Zhao
Keyword(s):  
Control Problem ◽  
Time Delays ◽  
Information Science ◽  
Networked Control Systems ◽  
Stability Result ◽  
Networked Control System ◽  
Networked Control ◽  
Packet Dropout ◽  
Interval Linear ◽  
The Stability

It is well known that stability of information transmission is very important. In this paper, we study the stability of Networked control systems. Networked control problem(NCSs) have received plentiful attention in recent two decades. A lot of researcher study the stability for NCSs. In this paper, we consider a class of interval linear system’s networked control problem with time delays and packet dropout. A stability result is obtained with the discrete-time method which is based on the Information Science.

scholarly journals Asymptotic Stability of the Golden-Section Control Law for Multi-Input and Multi-Output Linear Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Duo-Qing Sun ◽  
Zhu-Mei Sun
Keyword(s):  
Asymptotic Stability ◽  
Linear Systems ◽  
Golden Section ◽  
Stability Result ◽  
Least Square ◽  
Control Law ◽  
Phase System ◽  
Characteristic Model ◽  
Model Based ◽  
The Stability

This paper is concerned with the problem of the asymptotic stability of the characteristic model-based golden-section control law for multi-input and multi-output linear systems. First, by choosing a set of polynomial matrices of the objective function of the generalized least-square control, we prove that the control law of the generalized least square can become the characteristic model-based golden-section control law. Then, based on both the stability result of the generalized least-square control system and the stability theory of matrix polynomial, the asymptotic stability of the closed loop system for the characteristic model under the control of the golden-section control law is proved for minimum phase system.

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