Practical Stability of Caputo fractional differential equations by Lyapunov functions

In this paper the statement of initial value problems for fractional differential equations with noninstantaneous impulses is given. These equations are adequate models for phenomena that are characterized by impulsive actions starting at arbitrary fixed points and remaining active on finite time intervals. Strict stability properties of fractional differential equations with non-instantaneous impulses by the Lyapunov approach is studied. An appropriate definition (based on the Caputo fractional Dini derivative of a function) for the derivative of Lyapunov functions among the Caputo fractional differential equations with non-instantaneous impulses is presented. Comparison results using this definition and scalar fractional differential equations with non-instantaneous impulses are presented and sufficient conditions for strict stability and uniform strict stability are given. Examples are given to illustrate the theory.

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Stability with Initial Time Difference of Caputo Fractional Differential Equations by Lyapunov Functions

Zeitschrift für Analysis und ihre Anwendungen ◽

10.4171/zaa/1579 ◽

2017 ◽

Vol 36 (1) ◽

pp. 49-77 ◽

Cited By ~ 6

Author(s):

Ravi Agarwal ◽

Donal O'Regan ◽

Snezhana Hristova

Keyword(s):

Differential Equations ◽

Fractional Differential Equations ◽

Lyapunov Functions ◽

Time Difference ◽

Initial Time ◽

Caputo Fractional Differential Equations ◽

Fractional Differential

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Caputo Fractional Differential Equations with Non-Instantaneous Random Erlang Distributed Impulses

Fractal and Fractional ◽

10.3390/fractalfract3020028 ◽

2019 ◽

Vol 3 (2) ◽

pp. 28 ◽

Cited By ~ 1

Author(s):

Snezhana Hristova ◽

Krasimira Ivanova

Keyword(s):

Differential Equations ◽

Fractional Differential Equations ◽

Finite Time ◽

Lyapunov Functions ◽

Random Variable ◽

Time Intervals ◽

Caputo Fractional Differential Equations ◽

Fractional Differential ◽

Dini Derivatives ◽

Moment Exponential Stability

The p-moment exponential stability of non-instantaneous impulsive Caputo fractional differential equations is studied. The impulses occur at random moments and their action continues on finite time intervals with initially given lengths. The time between two consecutive moments of impulses is the Erlang distributed random variable. The study is based on Lyapunov functions. The fractional Dini derivatives are applied.

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Strict stability with respect to initial time difference for Caputo fractional differential equations by Lyapunov functions

Georgian Mathematical Journal ◽

10.1515/gmj-2016-0080 ◽

2017 ◽

Vol 24 (1) ◽

pp. 1-13 ◽

Cited By ~ 1

Author(s):

Ravi P. Agarwal ◽

Donal O’Regan ◽

Snezhana Hristova

Keyword(s):

Differential Equations ◽

Fractional Differential Equations ◽

Lyapunov Functions ◽

Sufficient Conditions ◽

Time Difference ◽

Initial Time ◽

Advantages And Disadvantages ◽

Strict Stability ◽

Caputo Fractional Differential Equations ◽

Fractional Differential

AbstractThe strict stability properties are generalized to nonlinear Caputo fractional differential equations in the case when both initial points and initial times are changeable. Using Lyapunov functions, some criteria for strict stability, eventually strict stability and strict practical stability are obtained. A brief overview of different types of derivatives in the literature related to the application of Lyapunov functions to Caputo fractional equations are given, and their advantages and disadvantages are discussed with several examples. The Caputo fractional Dini derivative with respect to to initial time difference is used to obtain some sufficient conditions.

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Lipschitz Stability for Non-Instantaneous Impulsive Caputo Fractional Differential Equations with State Dependent Delays

Axioms ◽

10.3390/axioms8010004 ◽

2018 ◽

Vol 8 (1) ◽

pp. 4 ◽

Cited By ~ 1

Author(s):

◽

Keyword(s):

Differential Equations ◽

Fractional Differential Equations ◽

Lyapunov Functions ◽

Sufficient Conditions ◽

Distributed Delay ◽

Lipschitz Stability ◽

Razumikhin Technique ◽

State Dependent ◽

Caputo Fractional Differential Equations ◽

Fractional Differential

In this paper, we study Lipschitz stability of Caputo fractional differential equations with non-instantaneous impulses and state dependent delays. The study is based on Lyapunov functions and the Razumikhin technique. Our equations in particular include constant delays, time variable delay, distributed delay, etc. We consider the case of impulses that start abruptly at some points and their actions continue on given finite intervals. The study of Lipschitz stability by Lyapunov functions requires appropriate derivatives among fractional differential equations. A brief overview of different types of derivative known in the literature is given. Some sufficient conditions for uniform Lipschitz stability and uniform global Lipschitz stability are obtained by an application of several types of derivatives of Lyapunov functions. Examples are given to illustrate the results.

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A survey of Lyapunov functions, stability and impulsive Caputo fractional differential equations

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2016-0017 ◽

2016 ◽

Vol 19 (2) ◽

Cited By ~ 33

Author(s):

Ravi Agarwal ◽

Snezhana Hristova ◽

Donal O’Regan

Keyword(s):

Differential Equations ◽

Fractional Differential Equations ◽

Lyapunov Functions ◽

Direct Method ◽

Sufficient Conditions ◽

Dini Derivative ◽

Advantages And Disadvantages ◽

Impulsive Fractional Differential Equations ◽

Caputo Fractional Differential Equations ◽

Fractional Differential

AbstractWe present an overview of the literature on solutions to impulsive Caputo fractional differential equations. Lyapunov direct method is used to obtain sufficient conditions for stability properties of the zero solution of nonlinear impulsive fractional differential equations. One of the main problems in the application of Lyapunov functions to fractional differential equations is an appropriate definition of its derivative among the differential equation of fractional order. A brief overview of those used in the literature is given, and we discuss their advantages and disadvantages. One type of derivative, the so called Caputo fractional Dini derivative, is generalized to impulsive fractional differential equations. We apply it to study stability and uniform stability. Some examples are given to illustrate the results.

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Applications of Lyapunov Functions to Caputo Fractional Differential Equations

Mathematics ◽

10.3390/math6110229 ◽

2018 ◽

Vol 6 (11) ◽

pp. 229

Author(s):

Ravi Agarwal ◽

Snezhana Hristova ◽

Donal O’Regan

Keyword(s):

Differential Equations ◽

Fractional Differential Equations ◽

Lyapunov Functions ◽

Sufficient Conditions ◽

Basic Question ◽

Caputo Fractional Differential Equations ◽

Fractional Differential ◽

Definition Of ◽

The Given ◽

Fractional Equation

One approach to study various stability properties of solutions of nonlinear Caputo fractional differential equations is based on using Lyapunov like functions. A basic question which arises is the definition of the derivative of the Lyapunov like function along the given fractional equation. In this paper, several definitions known in the literature for the derivative of Lyapunov functions among Caputo fractional differential equations are given. Applications and properties are discussed. Several sufficient conditions for stability, uniform stability and asymptotic stability with respect to part of the variables are established. Several examples are given to illustrate the theory.

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