Stability analysis of linear distributed order fractional systems with distributed delays

2017 ◽  
Vol 20 (4) ◽  
Author(s):  
Doychin Boyadzhiev ◽  
Hristo Kiskinov ◽  
Magdalena Veselinova ◽  
Andrey Zahariev
Keyword(s):  
Stability Analysis ◽  
Asymptotic Stability ◽  
Density Function ◽  
Global Asymptotic Stability ◽  
Fractional Derivatives ◽  
Sufficient Conditions ◽  
Zero Solution ◽  
Distributed Delays ◽  
Fractional Systems ◽  
Distributed Order

AbstractIn the present work we study linear systems with distributed delays and distributed order fractional derivatives based on Caputo type single fractional derivatives, with respect to a nonnegative density function. For the initial problem of this kind of systems, existence, uniqueness and a priory estimate of the solution are proved. As an application of the obtained results, we establish sufficient conditions for global asymptotic stability of the zero solution of the investigated types of systems.

Conditions for the local and global asymptotic stability of the time–fractional Degn–Harrison system

2020 ◽  
Vol 21 (7-8) ◽  
pp. 749-759
Author(s):  
Rachida Mezhoud ◽  
Khaled Saoudi ◽  
Abderrahmane Zaraï ◽  
Salem Abdelmalek
Keyword(s):  
Asymptotic Stability ◽  
Global Asymptotic Stability ◽  
Lyapunov Method ◽  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Linearization Method ◽  
Direct Lyapunov Method ◽  
Fractional Systems ◽  
Reaction Diffusion Model ◽  
Theoretical Results

AbstractFractional calculus has been shown to improve the dynamics of differential system models and provide a better understanding of their dynamics. This paper considers the time–fractional version of the Degn–Harrison reaction–diffusion model. Sufficient conditions are established for the local and global asymptotic stability of the model by means of invariant rectangles, the fundamental stability theory of fractional systems, the linearization method, and the direct Lyapunov method. Numerical simulation results are used to illustrate the theoretical results.

scholarly journals On uniform asymptotic stability for nonlinear integro-differential equations of Volterra type

2019 ◽  
pp. 161-166
Author(s):  
Natalia Sedova
Keyword(s):  
Stability Analysis ◽  
Asymptotic Stability ◽  
Sufficient Conditions ◽  
Zero Solution ◽  
Auxiliary Function ◽  
Uniform Asymptotic Stability ◽  
Volterra Type ◽  
Razumikhin Technique ◽  
The Stability ◽  
The Right

The specifics of the application of Razumikhin technique to the stability analysis of Volterra type integrodifferential equations are considered. The equation can be nonlinear and nonautonomous. We propose new sufficient conditions for uniform asymptotic stability of the zero solution using the phase space of a special construction and constraints on the right side of the equation. In the presented constraints we can analyze stability, relying not only on the behavior of the auxiliary function along the solutions, but also on the properties of the so called limiting equations.

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 THE GENERALISED LIÉNARD EQUATIONS

2009 ◽  
Vol 51 (3) ◽  
pp. 605-617
Author(s):  
A. AGHAJANI ◽  
A. MORADIFAM
Keyword(s):  
Asymptotic Stability ◽  
Periodic Solutions ◽  
Global Asymptotic Stability ◽  
Sufficient Conditions ◽  
Zero Solution ◽  
Oscillation Theory ◽  
Homoclinic Orbits ◽  
Liénard Equations ◽  
Existence Of Periodic Solutions ◽  
Image Position

AbstractIn this paper we present sufficient conditions for all trajectories of the system to cross the vertical isocline h(y) = F(x), which is very important in the global asymptotic stability of the origin, oscillation theory and existence of periodic solutions. Also we give sufficient conditions for all trajectories which start at a point on the curve h(y) = F(x), to cross the y-axis which is closely connected with the existence of homoclinic orbits, stability of the zero solution, oscillation theory and the centre problem. The obtained results extend and improve some of the authors' previous results and some other theorems in the literature.

Stability analysis of differential equations with maximum

2013 ◽  
Vol 63 (6) ◽  
Author(s):  
Ivanka Stamova ◽  
Gani Stamov
Keyword(s):  
Stability Analysis ◽  
Asymptotic Stability ◽  
Differential Equations ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Zero Solution ◽  
Uniform Stability

AbstractIn this paper, we investigate stability of the zero solution of differential equations with maximum by using Lyapunov functions and Razumikhin techniques. Sufficient conditions for stability, uniform stability and asymptotic stability of the zero solution of such equations are found.

scholarly journals On the Uniform Exponential Stability of Time-Varying Systems Subject to Discrete Time-Varying Delays and Nonlinear Delayed Perturbations

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Maher Hammami ◽  
Mohamed Ali Hammami ◽  
Manuel De la Sen
Keyword(s):  
Stability Analysis ◽  
Asymptotic Stability ◽  
Discrete Time ◽  
Global Asymptotic Stability ◽  
Linear Time ◽  
Sufficient Conditions ◽  
Time Varying ◽  
Uniform Exponential Stability ◽  
Time Varying Systems ◽  
Delayed Time

This paper addresses the problem of stability analysis of systems with delayed time-varying perturbations. Some sufficient conditions for a class of linear time-varying systems with nonlinear delayed perturbations are derived by using an improved Lyapunov-Krasovskii functional. The uniform global asymptotic stability of the solutions is obtained in terms of convergence toward a neighborhood of the origin.

scholarly journals Stability Analysis of Distributed Order Fractional Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
H. Saberi Najafi ◽  
A. Refahi Sheikhani ◽  
A. Ansari
Keyword(s):  
Stability Analysis ◽  
Characteristic Function ◽  
Differential Equations ◽  
Density Function ◽  
Robust Stability ◽  
Fractional Differential Equations ◽  
Stability Condition ◽  
Fractional Differential ◽  
The Stability ◽  
Distributed Order

We analyze the stability of three classes of distributed order fractional differential equations (DOFDEs) with respect to the nonnegative density function. In this sense, we discover a robust stability condition for these systems based on characteristic function and new inertia concept of a matrix with respect to the density function. Moreover, we check the stability of a distributed order fractional WINDMI system to illustrate the validity of proposed procedure.

On Stability of Solutions of Certain Differential Equations of the Third Order

1967 ◽  
Vol 10 (5) ◽  
pp. 681-688 ◽  
Author(s):  
B.S. Lalli
Keyword(s):  
Differential Equation ◽  
Asymptotic Stability ◽  
Lyapunov Function ◽  
Differential Equations ◽  
Global Asymptotic Stability ◽  
Sufficient Conditions ◽  
Stability Of Solutions ◽  
Third Order ◽  
The Third ◽  
Image Position

The purpose of this paper is to obtain a set of sufficient conditions for “global asymptotic stability” of the trivial solution x = 0 of the differential equation1.1using a Lyapunov function which is substantially different from similar functions used in [2], [3] and [4], for similar differential equations. The functions f1, f2 and f3 are real - valued and are smooth enough to ensure the existence of the solutions of (1.1) on [0, ∞). The dot indicates differentiation with respect to t. We are taking a and b to be some positive parameters.

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