scholarly journals Stability of a Functional Differential System with a Finite Number of Delays

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Josef Rebenda ◽  
Zdeněk Šmarda
Keyword(s):  
Asymptotic Stability ◽  
Differential System ◽  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Two Dimensional ◽  
Stability Of Solutions ◽  
Functional Differential ◽  
The Stability ◽  
Complex Valued ◽  
Functional Differential System

The paper is devoted to the study of asymptotic properties of a real two-dimensional differential system with unbounded nonconstant delays. The sufficient conditions for the stability and asymptotic stability of solutions are given. Used methods are based on the transformation of the considered real system to one equation with complex-valued coefficients. Asymptotic properties are studied by means of Lyapunov-Krasovskii functional. The results generalize some previous ones, where the asymptotic properties for two-dimensional systems with one or more constant delays or one nonconstant delay were studied.

scholarly journals Asymptotic Stability of the Solutions of Neutral Linear Fractional System with Nonlinear Perturbation

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 390
Author(s):  
Andrey Zahariev ◽  
Hristo Kiskinov
Keyword(s):  
Asymptotic Stability ◽  
Differential System ◽  
Global Asymptotic Stability ◽  
Linear Part ◽  
Sufficient Conditions ◽  
Zero Solution ◽  
Neutral System ◽  
Perturbed System ◽  
The Stability ◽  
Several Delays

In this article existence and uniqueness of the solutions of the initial problem for neutral nonlinear differential system with incommensurate order fractional derivatives in Caputo sense and with piecewise continuous initial function is proved. A formula for integral presentation of the general solution of a linear autonomous neutral system with several delays is established and used for the study of the stability properties of a neutral autonomous nonlinear perturbed linear fractional differential system. Natural sufficient conditions are found to ensure that from global asymptotic stability of the zero solution of the linear part of a nonlinearly perturbed system it follows global asymptotic stability of the zero solution of the whole nonlinearly perturbed system.

scholarly journals Stability and boundedness to certain differential equations of fourth order with multiple delays

Filomat ◽  
2014 ◽  
Vol 28 (5) ◽  
pp. 1049-1058 ◽  
Author(s):  
Erdal Korkmaz ◽  
Cemil Tunc
Keyword(s):  
Asymptotic Stability ◽  
Differential Equations ◽  
Fourth Order ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Functional Approach ◽  
Multiple Delays ◽  
Boundedness Of Solutions ◽  
Functional Differential ◽  
The Stability

In this paper, we give sufficient conditions to guarantee the asymptotic stability and boundedness of solutions to a kind of fourth-order functional differential equations with multiple delays. By using the Lyapunov-Krasovskii functional approach, we establish two new results on the stability and boundedness of solutions, which include and improve some related results in the literature.

scholarly journals On Stability of Linear Delay Differential Equations under Perron's Condition

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
J. Diblík ◽  
A. Zafer
Keyword(s):  
Asymptotic Stability ◽  
Differential Equations ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Zero Solution ◽  
Uniform Asymptotic Stability ◽  
Linear Delay ◽  
Functional Differential ◽  
The Stability ◽  
Delay Differential

The stability of the zero solution of a system of first-order linear functional differential equations with nonconstant delay is considered. Sufficient conditions for stability, uniform stability, asymptotic stability, and uniform asymptotic stability are established.

scholarly journals Asymptotic stability of the time-changed stochastic delay differential equations with Markovian switching

2021 ◽  
Vol 19 (1) ◽  
pp. 614-628
Author(s):  
Xiaozhi Zhang ◽  
Zhangsheng Zhu ◽  
Chenggui Yuan
Keyword(s):  
Asymptotic Stability ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Sufficient Conditions ◽  
Markovian Switching ◽  
Stability Of Solutions ◽  
Stochastic Delay Differential Equations ◽  
Stochastic Delay ◽  
The Stability ◽  
Delay Differential

Abstract The aim of this work is to study the asymptotic stability of the time-changed stochastic delay differential equations (SDDEs) with Markovian switching. Some sufficient conditions for the asymptotic stability of solutions to the time-changed SDDEs are presented. In contrast to the asymptotic stability in existing articles, we present the new results on the stability of solutions to time-changed SDDEs, which is driven by time-changed Brownian motion. Finally, an example is given to demonstrate the effectiveness of the main results.

ON THE STABILITY OF SOLUTIONS OF NONLINEAR SYSTEMS WITH IMPULSE STRUCTURE

2018 ◽  
pp. 624-636
Author(s):  
Natalia Igorevna Zhelonkina ◽  
Alexander Nikolaevich Sesekin
Keyword(s):  
Nonlinear Systems ◽  
Asymptotic Stability ◽  
Stability Property ◽  
Sufficient Conditions ◽  
Functions Of Bounded Variation ◽  
Stability Of Solutions ◽  
Smooth Solutions ◽  
Systems Of Differential Equations ◽  
The Stability ◽  
The Right

In this paper we review the results of the authors related to the study of the stability property of solutions for nonlinear systems of differential equations, on the right-hand side of which there are terms containing products of discontinuous functions and distributions. The solutions of such systems are formalized by the closure of the set of smooth solutions in the space of functions of bounded variation. For such systems, sufficient conditions are obtained for the asymptotic stability of unperturbed solutions.

Non-Oscillatory Solutions of Linear Delay Functional Differential System of Neutral Type

2012 ◽  
Vol 482-484 ◽  
pp. 66-69
Author(s):  
Xin Zhou Yan
Keyword(s):  
Differential System ◽  
Neutral Type ◽  
Sufficient Conditions ◽  
Oscillatory Solutions ◽  
Linear Delay ◽  
Functional Differential ◽  
Functional Differential System

In this paper, we consider the non-oscillatory solutions of linear delay functional differential system of neutral type, from which we obtain some new sufficient conditions of non-oscillation.

scholarly journals STABILITY OF ZERO SOLUTION OF SYSTEM WITH SWITCHES CONSISTING OF LINEAR SUBSYSTEMS

2020 ◽  
pp. 89-96
Author(s):  
D. Khusainov ◽  
A. Bychkov ◽  
A. Sirenko
Keyword(s):  
Asymptotic Stability ◽  
Lyapunov Function ◽  
Dynamic Systems ◽  
Sufficient Conditions ◽  
Zero Solution ◽  
Switching Systems ◽  
Stability Of Solutions ◽  
Quadratic Lyapunov Function ◽  
Linear Differential ◽  
The Stability

In this paper, discusses the study of the stability of solutions of dynamic systems with switching. Sufficient conditions are obtained for the asymptotic stability of the zero solution of switching systems consisting of linear differential and difference subsystems. It is proved that the existence of a common quadratic Lyapunov function is sufficient for asymptotic stability.

scholarly journals Asymptotic Behaviour of a Two-Dimensional Differential System with a Finite Number of Nonconstant Delays under the Conditions of Instability

2012 ◽  
Vol 2012 ◽  
pp. 1-20
Author(s):  
Zdeněk Šmarda ◽  
Josef Rebenda
Keyword(s):  
Finite Number ◽  
Asymptotic Behaviour ◽  
Vector Function ◽  
Differential System ◽  
Asymptotic Properties ◽  
Matrix Functions ◽  
Two Dimensional ◽  
Bounded Solutions ◽  
Complex Valued ◽  
Topological Principle

The asymptotic behaviour of a real two-dimensional differential system with unbounded nonconstant delays satisfying is studied under the assumption of instability. Here, , and are supposed to be matrix functions and a vector function. The conditions for the instable properties of solutions and the conditions for the existence of bounded solutions are given. The methods are based on the transformation of the considered real system to one equation with complex-valued coefficients. Asymptotic properties are studied by means of a Lyapunov-Krasovskii functional and the suitable Ważewski topological principle. The results generalize some previous ones, where the asymptotic properties for two-dimensional systems with one constant or nonconstant delay were studied.

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