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

2017 ◽  
Vol 36 (1) ◽  
pp. 49-77 ◽  
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

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

2017 ◽  
Vol 24 (1) ◽  
pp. 1-13 ◽  
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.

Some stability properties related to initial time difference for Caputo fractional differential equations

2018 ◽  
Vol 21 (1) ◽  
pp. 72-93 ◽  
Author(s):  
Ravi Agarwal ◽  
Snezhana Hristova ◽  
Donal O’Regan
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Sufficient Conditions ◽  
Time Difference ◽  
Initial Time ◽  
Lipschitz Stability ◽  
Caputo Fractional Differential Equations ◽  
Fractional Differential ◽  
Order Extension

Abstract Lipschitz stability and Mittag-Leffler stability with initial time difference for nonlinear nonautonomous Caputo fractional differential equation are defined and studied using Lyapunov like functions. Some sufficient conditions are obtained. The fractional order extension of comparison principles via scalar fractional differential equations with a parameter is employed. The relation between both types of stability is discussed theoretically and it is illustrated with examples.

Caputo fractional differential equations with non-instantaneous impulses and strict stability by Lyapunov functions

Filomat ◽  
2017 ◽  
Vol 31 (16) ◽  
pp. 5217-5239 ◽  
Author(s):  
Ravi Agarwal ◽  
Snehana Hristova ◽  
Donal O’Regan
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Time Intervals ◽  
Dini Derivative ◽  
Comparison Results ◽  
Strict Stability ◽  
Caputo Fractional Differential Equations ◽  
Fractional Differential

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.

scholarly journals Lyapunov Functions and Lipschitz Stability for Riemann–Liouville Non-Instantaneous Impulsive Fractional Differential Equations

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 730
Author(s):  
Ravi Agarwal ◽  
Snezhana Hristova ◽  
Donal O’Regan
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Initial Time ◽  
Lipschitz Stability ◽  
Comparison Results ◽  
Liouville Fractional Derivative ◽  
Fractional Differential ◽  
Definition Of

In this paper a system of nonlinear Riemann–Liouville fractional differential equations with non-instantaneous impulses is studied. We consider a Riemann–Liouville fractional derivative with a changeable lower limit at each stop point of the action of the impulses. In this case the solution has a singularity at the initial time and any stop time point of the impulses. This leads to an appropriate definition of both the initial condition and the non-instantaneous impulsive conditions. A generalization of the classical Lipschitz stability is defined and studied for the given system. Two types of derivatives of the applied Lyapunov functions among the Riemann–Liouville fractional differential equations with non-instantaneous impulses are applied. Several sufficient conditions for the defined stability are obtained. Some comparison results are obtained. Several examples illustrate the theoretical results.

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

2019 ◽  
Vol 3 (2) ◽  
pp. 28 ◽  
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.

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

Axioms ◽  
2018 ◽  
Vol 8 (1) ◽  
pp. 4 ◽  
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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