Global Mittag—Leffler Synchronization for Neural Networks Modeled by Impulsive Caputo Fractional Differential Equations with Distributed Delays
Abstract
:1. Introduction
2. Impulses in Fractional Delay Differential Equations
3. Lyapunov Functions and Their Fractional Derivatives
- 1.
- The function is continuous on and it is locally Lipschitz with respect to its second argument;
- 2.
- For each and , there exist finite limits
- -
- first type– Let be a solution of the IVP for the IFrDDE (3) (according to A2 for IFrDDE). Then, we can consider the Caputo fractional derivative of the function defined byThis type of derivative is applicable for continuously differentiable Lyapunov functions.
- -
- second type– Let Then, the Dini fractional derivative of the Lyapunov function is defined byThe Dini fractional derivative (8) keeps the concept of fractional derivatives because it has a memory.Note that Dini fractional derivative, defined by (8), is based on the notationIn [17], the notation (9) is used directly. However, the notation (9) does not depend on the order q of the fractional derivative nor on the initial time , which is typical for the Caputo fractional derivative. The operator defined by (9) has no memory. In addition, if is a solution of (3), then .For fractional differential equations without any impulses, notation similar to (9) is defined and is applied [16].
- -
- third type—let the initial data of IVP for IFrDDE (3) and be given. Then, the Caputo fractional Dini derivative of the Lyapunov function is defined by:or its equivalentThe derivative given by (11) depends significantly on both the fractional order q and the initial data of IVP for IFrDDE (3) and it makes this type of derivative close to the idea of the Caputo fractional derivative of a function.
4. Some Comparison Results for Lyapunov Functions
4.1. Comparison Results for Delay Fractional Differential Equations
- 1.
- The function is a solution of the IVP for FrDDE (17).
- 2.
- The function , is such that there exist positive numbers such that, for any point , the fractional derivative exists and the inequality
4.2. Comparison Results for Impulsive Delay Fractional Differential Equations
- 1.
- The function is a solution of the IVP for FrDDE (3) with .
- 2.
- The function , is such that there exist positive numbers :
- (i)
- for any and any point such that , the fractional derivative exists and the inequality
- (ii)
- for all and , the inequality .
- (i)
- for any and any point and any function , the inequality
- (ii)
- for all and the inequality .
5. Application to Neural Networks
5.1. Problem Formulation
5.2. Mittag–Leffler Synchronization
5.2.1. Output Coupling Controller
5.2.2. State Coupling Controllers
6. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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Agarwal, R.; Hristova, S.; O’Regan, D. Global Mittag—Leffler Synchronization for Neural Networks Modeled by Impulsive Caputo Fractional Differential Equations with Distributed Delays. Symmetry 2018, 10, 473. https://doi.org/10.3390/sym10100473
Agarwal R, Hristova S, O’Regan D. Global Mittag—Leffler Synchronization for Neural Networks Modeled by Impulsive Caputo Fractional Differential Equations with Distributed Delays. Symmetry. 2018; 10(10):473. https://doi.org/10.3390/sym10100473
Chicago/Turabian StyleAgarwal, Ravi, Snezhana Hristova, and Donal O’Regan. 2018. "Global Mittag—Leffler Synchronization for Neural Networks Modeled by Impulsive Caputo Fractional Differential Equations with Distributed Delays" Symmetry 10, no. 10: 473. https://doi.org/10.3390/sym10100473