scholarly journals Semi-Definite Lyapunov Functionals in the Stability Problem of Volterra Integral-Differential Equations

2019 ◽  
Vol 52 (18) ◽  
pp. 103-108
Author(s):  
Aleksandr S. Andreev ◽  
Olga A. Peregudova
Keyword(s):  
Differential Equations ◽  
Stability Problem ◽  
Lyapunov Functionals ◽  
Integral Differential Equations ◽  
The Stability

scholarly journals On the Lyapunov functionals method in the stability problem of Volterra integro-differential equations

2018 ◽  
Vol 20 (3) ◽  
pp. 260-272
Author(s):  
A. S. Andreev ◽  
O. A. Peregudova
Keyword(s):  
Differential Equations ◽  
Stability Problem ◽  
Infinite Delay ◽  
Zero Solution ◽  
Model Systems ◽  
Lyapunov Functionals ◽  
Structure Equations ◽  
Positive Limit Set ◽  
The Stability ◽  
The Right

In this paper, we consider the problem of applying the method of Lyapunov functionals to investigate the stability of non-linear integro-differential equations, the right-hand side of which is the sum of the components of the instantaneous action and also ones with a finite and infinite delay. The relevance of the problem is the widespread use of such complicated in structure equations in modeling the controllers using integral regulators for mechanical systems, as well as biological, physical and other processes. We develop the Lyapunov functionals method in the direction of revealing the limiting properties of solutions by means of Lyapunov functionals with a semi-definite derivative. We proved the theorems on the quasi-invariance of a positive limit set of bounded solution as well as ones on the asymptotic stability of the zero solution including a uniform one. The results are achieved by constructing a new structure of the topological dynamics of the equations under study. The theorems proved are applied in solving the stability problem of two model systems which are generalizations of a number of known models of natural science and technology.

scholarly journals On the Lyapunov functionals method in the stability problem of Volterra integro-differential equations

2018 ◽  
Vol 20 (3) ◽  
pp. 260-272
Author(s):  
Aleksandr S. Andreev ◽  
Olga A. Peregudova
Keyword(s):  
Differential Equations ◽  
Stability Problem ◽  
Infinite Delay ◽  
Zero Solution ◽  
Model Systems ◽  
Lyapunov Functionals ◽  
Structure Equations ◽  
Positive Limit Set ◽  
The Stability ◽  
The Right

In this paper, we consider the problem of applying the method of Lyapunov functionals to investigate the stability of non-linear integro-differential equations, the right-hand side of which is the sum of the components of the instantaneous action and also ones with a finite and infinite delay. The relevance of the problem is the widespread use of such complicated in structure equations in modeling the controllers using integral regulators for mechanical systems, as well as biological, physical and other processes. We develop the Lyapunov functionals method in the direction of revealing the limiting properties of solutions by means of Lyapunov functionals with a semi-definite derivative. We proved the theorems on the quasi-invariance of a positive limit set of bounded solution as well as ones on the asymptotic stability of the zero solution including a uniform one. The results are achieved by constructing a new structure of the topological dynamics of the equations under study. The theorems proved are applied in solving the stability problem of two model systems which are generalizations of a number of known models of natural science and technology.

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.

On the Stability Problem of Differential Equations in the Sense of Ulam

2019 ◽  
Vol 75 (1) ◽  
Author(s):  
Yasemin Başcı ◽  
Adil Mısır ◽  
Süleyman Öğrekçi
Keyword(s):  
Differential Equations ◽  
Stability Problem ◽  
The Stability

scholarly journals Application of M-Matrices for the Study of Mathematical Models of Living Systems

2018 ◽  
Vol 13 (1) ◽  
pp. 208-237 ◽  
Author(s):  
N.V. Pertsev ◽  
B.Yu. Pichugin ◽  
A.N. Pichugina
Keyword(s):  
Differential Equations ◽  
Stability Problem ◽  
Distributed Delay ◽  
Linearization Method ◽  
Living Systems ◽  
High Dimensional ◽  
Linear Differential ◽  
The Stability ◽  
Several Delays ◽  
The Right

Some results are presented of application of M-matrices to the study the stability problem of the equilibriums of differential equations used in models of living systems. The models studied are described by differential equations with several delays, including distributed delay, and by high-dimensional systems of differential equations. To study the stability of the equilibriums the linearization method is used. Emerging systems of linear differential equations have a specific structure of the right-hand parts, which allows to effectively use the properties of M-matrices. As examples, the results of studies of models arising in immunology, epidemiology and ecology are presented.

scholarly journals Uniform Stability In Nonlinear Infinite Delay Volterra Integro-differential Equations Using Lyapunov Functionals

2016 ◽  
Vol 3 (1) ◽  
Author(s):  
Youssef Raffoul ◽  
Habib Rai
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Exponential Stability ◽  
Lyapunov Functional ◽  
Infinite Delay ◽  
Zero Solution ◽  
Uniform Stability ◽  
Lyapunov Functionals ◽  
Integro Differential Equation ◽  
The Stability

AbstractIn [10] the first author used Lyapunov functionals and studied the exponential stability of the zero solution of finite delay Volterra Integro-differential equation. In this paper, we use modified version of the Lyapunov functional that were used in [10] to obtain criterion for the stability of the zero solution of the infinite delay nonlinear Volterra integro-differential equation

scholarly journals A Generalization of Bihari's Lemma for Discontinuous Functions and Its Application to the Stability Problem of Differential Equations with Impulse Disturbance

2006 ◽  
Vol 13 (2) ◽  
pp. 229-238
Author(s):  
Sergiy Borysenko ◽  
Giovanni Matarazzo ◽  
Massimo Pecoraro
Keyword(s):  
Differential Equations ◽  
Stability Problem ◽  
Integral Inequalities ◽  
Practical Stability ◽  
Stability Of Solutions ◽  
Discontinuous Functions ◽  
The Stability

Abstract This paper presents a generalization of nonlinear integral inequalities of the Gronwall–Bellman–Bihari type for discontinuous functions and its application to the investigation of the practical stability of solutions of systems of integro-differential equations with impulse perturbations at fixed moments of time.

scholarly journals Stability of Perturbed Set Differential Equations Involving Causal Operators in Regard to Their Unperturbed Ones considering Difference in Initial Conditions

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Coșkun Yakar ◽  
Hazm Talab
Keyword(s):  
Differential Equations ◽  
Lyapunov Functions ◽  
Initial Conditions ◽  
Lyapunov Functionals ◽  
Stability Of Solutions ◽  
Causal Operators ◽  
The Difference ◽  
The Stability

We investigate the stability of solutions of perturbed set differential equations with causal operators in regard to their corresponding unperturbed ones considering the difference in initial conditions (time and position) by utilizing Lyapunov functions and Lyapunov functionals.

Sign in / Sign up

Export Citation Format

Share Document