scholarly journals Second method of Lyapunov for stability of linear impulsive differential-difference equations with variable impulsive perturbations

1998 ◽  
Vol 11 (2) ◽  
pp. 209-216 ◽  
Author(s):  
D. D. Bainov ◽  
I. M. Stamova ◽  
A. S. Vatsala
Keyword(s):  
Difference Equations ◽  
Sufficient Conditions ◽  
Zero Solution ◽  
Asymptotical Stability ◽  
Uniform Stability ◽  
Auxiliary Functions ◽  
Piecewise Continuous ◽  
Impulsive Perturbations ◽  
Second Method Of Lyapunov

The present work is devoted to the study of stability of the zero solution to linear impulsive differential-difference equations with variable impulsive perturbations. With the aid of piecewise continuous auxiliary functions, which are generalizations of the classical Lyapunov's functions, sufficient conditions are found for the uniform stability and uniform asymptotical stability of the zero solution to equations under consideration.

scholarly journals Vector Lyapunov functions and conditional stability for systems of impulsive differential-difference equations

2001 ◽  
Vol 42 (3) ◽  
pp. 341-353 ◽  
Author(s):  
D. D. Bainov ◽  
I. M. Stamova
Keyword(s):  
Difference Equations ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Zero Solution ◽  
Comparison Method ◽  
Conditional Stability ◽  
Continuous Vector ◽  
Vector Lyapunov Functions ◽  
Piecewise Continuous

AbstractBy means of piecewise continuous vector functions, which are analogues of the classical Lyapunov functions and via the comparison method, sufficient conditions are found for conditional, stability of the zero solution of a system of impulsive differential-difference equations.

An Asymptotical Stability in Variational Sense for Second Order Systems

2010 ◽  
Vol 10 (1) ◽  
Author(s):  
Dariusz Idczak ◽  
Stanisław Walczak
Keyword(s):  
Sobolev Space ◽  
Differential System ◽  
Sufficient Conditions ◽  
Zero Solution ◽  
Second Order ◽  
Functional Parameter ◽  
Asymptotical Stability ◽  
Parameter Control ◽  
Initial Condition ◽  
Second Order Systems

AbstractIn this paper, a new, variational concept of asymptotical stability of zero solution to an ordinary differential system of the second order, considered in Sobolev space, is presented. Sufficient conditions for an asymptotical stability in a variational sense with respect to initial condition and functional parameter (control) are given. Relation to the classical asymptotical stability is illustrated.

GLOBAL ATTRACTIVITY IN A DELAY POPULATION MODEL UNDER IMPULSIVE PERTURBATIONS

2002 ◽  
Vol 34 (3) ◽  
pp. 319-328 ◽  
Author(s):  
J. S. YU ◽  
X. H. TANG
Keyword(s):  
Population Model ◽  
Global Attractivity ◽  
Sufficient Conditions ◽  
Zero Solution ◽  
Impulsive Perturbations

This paper provides sufficient conditions that guarantee the global attractivity of the zero solution of a delay population model under impulsive perturbations.

scholarly journals Existence and Global Asymptotical Stability of Periodic Solution for theT-Periodic Logistic System with Time-Varying Generating Operators andT0-Periodic Impulsive Perturbations on Banach Spaces

2008 ◽  
Vol 2008 ◽  
pp. 1-16 ◽  
Author(s):  
JinRong Wang ◽  
X. Xiang ◽  
W. Wei ◽  
Qian Chen
Keyword(s):  
Banach Spaces ◽  
Mild Solution ◽  
Sufficient Conditions ◽  
Asymptotical Stability ◽  
Time Varying ◽  
Global Asymptotical Stability ◽  
Globally Asymptotically Stable ◽  
Impulsive Perturbations ◽  
Generating Operators ◽  
Well Posed

This paper studies the existence and global asymptotical stability of periodic PC-mild solution for theT-periodic Logistic system with time-varying generating operators andT0-periodic impulsive perturbations on Banach spaces. Two sufficient conditions that guarantee the exponential stability of the impulsive evolution operator corresponding to homogenous well-posedT-periodic system with time-varying generating operators andT0-periodic impulsive perturbations are given. It is shown that the system have a unique periodic PC-mild solution which is globally asymptotically stable whenTandT0are rational dependent and its period must benT0for somen∈N. At last, an example is given for demonstration.

scholarly journals Necessary and sufficient conditions for uniform stability of Volterra integro-dynamic equations using new resolvent equation

2013 ◽  
Vol 21 (3) ◽  
pp. 17-32 ◽  
Author(s):  
Murat Advar ◽  
Youssef N. Raffoul
Keyword(s):  
Lyapunov Functional ◽  
Sufficient Conditions ◽  
Zero Solution ◽  
Necessary And Sufficient Conditions ◽  
Uniform Stability ◽  
Dynamic Equations ◽  
Resolvent Equation ◽  
Variation Of Parameters ◽  
Variation Of Parameters Formula ◽  
Necessary And Sufficient

AbstractWe consider the system of Volterra integro-dynamic equations and obtain necessary and sufficient conditions for the uniform stability of the zero solution employing the resolvent equation coupled with the variation of parameters formula. The resolvent equation that we use for the study of stability will have to be developed since it is unknown for time scales. At the end of the paper, we furnish an example in which we deploy an appropriate Lyapunov functional. In addition to generalization, the results of this paper provides improvements for its counterparts in integro-differential and integro-difference equations which are the most important particular cases of our equation.

scholarly journals Second method of Lyapunov and comparison principle for impulsive differential–difference equations

1997 ◽  
Vol 38 (4) ◽  
pp. 489-505 ◽  
Author(s):  
D. D. Bainov ◽  
I. M. Stamova
Keyword(s):  
Comparison Principle ◽  
Difference Equations ◽  
Continuous Functions ◽  
Differential Inequalities ◽  
Impulse Effect ◽  
Comparison Equation ◽  
Piecewise Continuous ◽  
Second Method Of Lyapunov

AbstractIn the present paper questions related to stability and boundedness with respect to manifolds of solutions of impulsive differential-difference equations are considered. The investigations are carried out by means of piecewise-continuous functions which are analogues of the classical Lyapunov's functions. By means of a vectorial comparison equation and differential inequalities for piecewise-continuous functions, theorems are proved on stability and boundedness with respect to manifolds of solutions of impulsive differential-difference equations with impulse effect at fixed moments.

scholarly journals BOUNDEDNESS OF IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS WITH VARIABLE IMPULSIVE PERTURBATIONS

2008 ◽  
Vol 77 (2) ◽  
pp. 331-345 ◽  
Author(s):  
I. M. Stamova
Keyword(s):  
Differential Equations ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Boundedness Of Solutions ◽  
Initial Value ◽  
Razumikhin Technique ◽  
Piecewise Continuous ◽  
Impulsive Perturbations ◽  
Functional Differential

AbstractIn the present paper an initial value problem for impulsive functional differential equations with variable impulsive perturbations is considered. By means of piecewise continuous functions coupled with the Razumikhin technique, sufficient conditions for boundedness of solutions of such equations are found.

scholarly journals Asymptotic stability of systems with impulses by the direct method of Lyapunov

1988 ◽  
Vol 38 (1) ◽  
pp. 113-123 ◽  
Author(s):  
G.K. Kulev ◽  
D.D. Bainov
Keyword(s):  
Asymptotic Stability ◽  
Direct Method ◽  
Zero Solution ◽  
Globally Asymptotic Stability ◽  
Auxiliary Functions ◽  
Piecewise Continuous

In the present paper the asymptotic and globally asymptotic stability of the zero solution of systems with impulses are investigated. For this purpose piecewise continuous auxiliary functions are used which are analogous to Lyapunov's functions. The theorem of Marachkov on the asymptotic stability of systems without impulses is generalised. The results obtained are formulated in four theorems.

scholarly journals Stability of the solutions of impulsive functional-differential equations by Lyapunov's direct method

2001 ◽  
Vol 43 (2) ◽  
pp. 269-278 ◽  
Author(s):  
D. D. Bainov ◽  
I. M. Stamova
Keyword(s):  
Differential Equations ◽  
Lyapunov Functions ◽  
Direct Method ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Zero Solution ◽  
Continuous Functions ◽  
Piecewise Continuous ◽  
Functional Differential ◽  
The Stability

AbstractWe consider the stability of the zero solution of a system of impulsive functional-differential equations. By means of piecewise continuous functions, which are generalizations of classical Lyapunov functions, and using a technique due to Razumikhin, sufficient conditions are found for stability, uniform stability and asymptotical stability of the zero solution of these equations. Applications to impulsive population dynamics are also discussed.

scholarly journals Stability conditions for linear delay difference equations: a survey and perspectives

2015 ◽  
Vol 63 (1) ◽  
pp. 1-29 ◽  
Author(s):  
Jan Čermák
Keyword(s):  
Asymptotic Stability ◽  
Difference Equations ◽  
Sufficient Conditions ◽  
Zero Solution ◽  
Stability Conditions ◽  
Linear Delay ◽  
Necessary And Sufficient ◽  
Delay Difference Equations ◽  
Delay Difference ◽  
Theoretical Results

Abstract The paper presents an overview of the basic results and methods for stability investigations of higher-order linear autonomous difference equations. The presented criteria formulate several types of necessary and sufficient conditions for the asymptotic stability of the zero solution of studied equations, with a special emphasize put on delay difference equations. Various comments, comparisons, examples and illustrations are given to support theoretical results.

Sign in / Sign up

Export Citation Format

Share Document