scholarly journals On the Stability of Nonautonomous Linear Impulsive Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
JinRong Wang ◽  
Xuezhu Li
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Impulsive Differential Equations ◽  
The Stability ◽  
Ulam’S Type Stability ◽  
Unbounded Intervals ◽  
Stability Results ◽  
Rassias Stability ◽  
Stability Concepts

We introduce two Ulam's type stability concepts for nonautonomous linear impulsive ordinary differential equations. Ulam-Hyers and Ulam-Hyers-Rassias stability results on compact and unbounded intervals are presented, respectively.

Stability of Solutions of Ordinary Differential Equations with Respect to a Closed Set

1968 ◽  
Vol 20 ◽  
pp. 720-726
Author(s):  
T. G. Hallam ◽  
V. Komkov
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Compact Set ◽  
Stability Of Solutions ◽  
Closed Set ◽  
The Stability ◽  
Stability Results

The stability of the solutions of an ordinary differential equation will be discussed here. The purpose of this note is to compare the stability results which are valid with respect to a compact set and the stability results valid with respect to an unbounded set. The stability of sets is a generalization of stability in the sense of Liapunov and has been discussed by LaSalle (5; 6), LaSalle and Lefschetz (7, p. 58), and Yoshizawa (8; 9; 10).

Stability results for fractional differential equations with state-dependent delay and not instantaneous impulses

2017 ◽  
Vol 67 (4) ◽  
Author(s):  
Saïd Abbas ◽  
Mouffak Benchohra
Keyword(s):  
Differential Equations ◽  
Functional Differential Equations ◽  
Darboux Problem ◽  
State Dependent ◽  
State Dependent Delay ◽  
Fractional Differential ◽  
Functional Differential ◽  
Ulam’S Type Stability ◽  
Stability Results ◽  
Stability Concepts

AbstractIn this paper, we shall present some uniqueness and Ulam’s type stability concepts for the Darboux problem of partial functional differential equations with not instantaneous impulses and state-dependent delay in Banach spaces. Some examples are also provided to illustrate our results.

scholarly journals Stability and solvability for a class of optimal control problems described by non-instantaneous impulsive differential equations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Yi Chen ◽  
Kaixuan Meng
Keyword(s):  
Optimal Control ◽  
Differential Equations ◽  
Optimal Control Problems ◽  
Impulsive Differential Equations ◽  
State Equation ◽  
Control Problems ◽  
Existence And Stability ◽  
Control Set ◽  
The Stability ◽  
Stability Results

Abstract In this paper, we investigate the existence and stability of solutions for a class of optimal control problems with 1-mean equicontinuous controls, and the corresponding state equation is described by non-instantaneous impulsive differential equations. The existence theorem is obtained by the method of minimizing sequence, and the stability results are established by using the related conclusions of set-valued mappings in a suitable metric space. An example with the measurable admissible control set, in which the controls are not continuous, is given in the end.

scholarly journals Practical Stability in terms of Two Measures for Impulsive Differential Equations with “Supremum”

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
S. G. Hristova ◽  
A. Georgieva
Keyword(s):  
Differential Equations ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Practical Stability ◽  
Time Interval ◽  
Comparison Results ◽  
Two Measures ◽  
The Stability ◽  
The Given ◽  
Stability Concepts

The object of investigations is a system of impulsive differential equations with “supremum.” These equations are not widely studied yet, and at the same time they are adequate mathematical model of many real world processes in which the present state depends significantly on its maximal value on a past time interval. Practical stability for a nonlinear system of impulsive differential equations with “supremum” is defined and studied. It is applied Razumikhin method with piecewise continuous scalar Lyapunov functions and comparison results for scalar impulsive differential equations. To unify a variety of stability concepts and to offer a general framework for the investigation of the stability theory, the notion of stability in terms of two measures has been applied to both the given system and the comparison scalar equation. An example illustrates the usefulness of the obtained sufficient conditions.

scholarly journals Stability Analysis of Multistage One-Step Multiderivative Methods for Nonlinear Impulsive Differential Equations in Banach Spaces

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Tao Long ◽  
Yuexin Yu
Keyword(s):  
Asymptotic Stability ◽  
Differential Equations ◽  
Banach Spaces ◽  
Impulsive Differential Equations ◽  
Problem Class ◽  
One Step ◽  
The Stability ◽  
Stability Results ◽  
Theoretical Results ◽  
Multiderivative Methods

In this paper, we first introduce the problem class K p μ , λ , ζ with respect to the initial value problems of nonlinear impulsive differential equations in Banach spaces. The stability and asymptotic stability results of the analytic solution of the problem class K p μ , λ , ζ are obtained. Then, the numerical stability and asymptotic stability conditions of multistage one-step multiderivative methods are also given. Two numerical experiments are given to confirm the theoretical results in the end.

scholarly journals Stability of delay differential equations in the sense of Ulam on unbounded intervals

2019 ◽  
Vol 9 (2) ◽  
pp. 125-131 ◽  
Author(s):  
Süleyman Öğrekçi
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Stability Problem ◽  
The Stability ◽  
Delay Differential ◽  
Unbounded Intervals

In this paper, we consider the stability problem of delay differential equations in the sense of Hyers-Ulam-Rassias. Recently this problem has been solved for bounded intervals, our result extends and improve the literature by obtaining stability in unbounded intervals. An illustrative example is also given to compare these results and visualize the improvement.

Stability in terms of two measures for population growth models with impulsive perturbations

2020 ◽  
Vol 13 (06) ◽  
pp. 2050051
Author(s):  
Zhinan Xia ◽  
Qianlian Wu ◽  
Dingjiang Wang
Keyword(s):  
Differential Equations ◽  
Population Growth ◽  
Ordinary Differential Equations ◽  
Trivial Solution ◽  
Growth Models ◽  
Two Measures ◽  
Impulsive Perturbations ◽  
The Stability ◽  
Generalized Ordinary Differential Equations ◽  
Population Growth Models

In this paper, we establish some criteria for the stability of trivial solution of population growth models with impulsive perturbations. The working tools are based on the theory of generalized ordinary differential equations. Here, the conditions concerning the functions are more general than the classical ones.

scholarly journals Ultimate Boundedness of the Systems Governed by Stochastic Differential Equations

1972 ◽  
Vol 47 ◽  
pp. 111-144 ◽  
Author(s):  
Yoshio Miyahara
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Stochastic Differential Equations ◽  
Functional Differential Equations ◽  
Ultimate Boundedness ◽  
Functional Differential ◽  
The Stability

The stability of the systems given by ordinary differential equations or functional-differential equations has been studied by many mathematicians. The most powerful tool in this field seems to be the Liapunov’s second method (see, for example [6]).

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