Global Mittag—Leffler Synchronization for Neural Networks Modeled by Impulsive Caputo Fractional Differential Equations with Distributed Delays

The synchronization problem for impulsive fractional-order neural networks with both time-varying bounded and distributed delays is studied. We study the case when the neural networks and the fractional derivatives of all neurons depend significantly on the moments of impulses and we consider both the cases of state coupling controllers and output coupling controllers. The fractional generalization of the Razumikhin method and Lyapunov functions is applied. Initially, a brief overview of the basic fractional derivatives of Lyapunov functions used in the literature is given. Some sufficient conditions are derived to realize the global Mittag–Leffler synchronization of impulsive fractional-order neural networks. Our results are illustrated with examples.


Introduction
Over the last few decades, fractional differential equations have gained considerable importance and attention due to their applications in science and engineering, i.e., in control, in stellar interiors, star clusters [1], in electrochemistry, in viscoelasticity [2] and in optics [3].For example, the control of mechanical systems is currently one of the most active fields of research and the use of fractional order calculus increases the flexibility of controlling any system from a point to a space.Applications of fractional quantum mechanics cover dynamics of a free particle and a new representation for a free particle quantum mechanical kernel (see, for example, [4]).
The stability of fractional order systems is quite a recent topic (see, for example, Ref. [5] for the Ulam-Hyers-Mittag-Leffler stability of fractional-order delay differential equations, Ref. [6] for the Mittag-Leffler stability of impulsive fractional neural network, Ref. [7] for the Mittag-Leffler stability of fractional systems, Ref. [8] for the Mittag-Leffler stability for fractional nonlinear systems with delay, and Ref. [9] for the Mittag-Leffler stability of nonlinear fractional systems with impulses).One of the most useful approaches in studying stability for nonlinear fractional differential equations is the Lyapunov approach.Its application to fractional differential equations is connected with several difficulties.One of the main difficulties is connected with the appropriate definition of derivative of Lyapunov functions among thedifferential equations of fractional order.Impulsive differential equations arise in real world problems to describe the dynamics of processes in which sudden, discontinuous jumps occur.
Most research on the synchronization of delayed neural networks has been restricted to the case of discrete delays (see, for example, [10]) Since a neural network usually has a spatial nature due to the presence of an amount of parallel pathways of a variety of axis sizes and lengths, it is desirable to model them by introducing distributed delays.Note in [11] that both time-varying delays and distributed time delays are taken into account in studying fractional neural networks with impulses and constant strengths between two units.In all models of neural networks, one considers the case of constant rate with which the i-th neuron resets its potential to the resting state in isolation, and the constant synaptic connection strength of the i-th neuron to the j-th neuron (see, for example, [10]).In our paper, we consider the general case of time varying coefficients in the model that allows more appropriate modeling of the connections between the neurons.These more complicated mathematical equations lead to an application of new types of fractional derivatives of Lyapunov functions and new stability results.
In this paper, Caputo fractional delay differential equations with impulses and two types of delays-variable in time and distributed ones are studied.Some results for piecewise continuous Lyapunov functions based on the Razumikhin method are obtained.Appropriate derivatives of Lyapunov functions among the studied fractional equations are used.Our results are applied to study the synchronization of neural networks with Caputo fractional derivatives, variable delays, distributed delays, and impulses.We study the case when the lower limit of the fractional derivative is changing after each impulsive time.To the best of our knowledge, this is the first model of neural networks of this type studied in the literature.Additionally, we study the general case of variables in time strengths of the j-th unit on the i-th unit and nonlinear impulsive functions.Both the cases of state coupling controllers and output coupling controllers are considered.Our sufficient conditions naturally depend significantly on the fractional order of the model (compare with sufficient conditions in [11,12]).
In many applications in science and engineering, the fractional order q is often less than 1, so we restrict q ∈ (0, 1) everywhere in the paper.

3:
The Grünwald-Letnikov fractional derivative is given by (see, for example, [13][14][15])) The Mittag-Leffler function with one parameter is defined as exists) and the Caputo derivative C t 0 D q m(t) exists for t ∈ [t 0 , T].Consider the initial value problem (IVP) for the nonlinear impulsive Caputo fractional delay differential equation (IFrDDE) We will denote the solution of the IVP for IFrDDE (1) by x(t; t 0 , φ) for t ≥ t 0 .The solution of IFrDDE ( 1) is a piecewise continuous function.In connection with this, we introduce the following sets of functions: The fractional derivatives depend significantly on their lower limit and it allows different interpretations of piecewise continuous solutions of impulsive differential equations.This phenomena is not characteristic for ordinary derivatives.In the literature, there are two main approaches to interpret the solutions of impulsive fractional delay differential equations: First approach to the solutions of (1) (A1 for IFrDDE).
The solution of the IVP for IFrDDE (1) satisfies the equalities (integral) Formula ( 2) is given and used in [17].It is a generalization to the formula proved in [18] for the solution of impulsive fractional differential equations without delays.
Second approach to the solutions of (1) (A2 for IFrDDE).
The idea of this approach is based on the dependence of the Caputo fractional derivative on the initial time point of the interval of differential equation, i.e., the lower limit of the Caputo fractional derivative is changing at each moment of impulse of the differential equation.Sometimes, Equation (1) in this case is written by Then, the solution of the IVP for IFrDDE (1), respectively (3), is given by Remark 1.Both Formulas (2) and ( 4) differ for fractional differential equations and they are generalizations to impulsive ordinary differential equations.Both formulas coincide in the case of the ordinary derivative (q = 1) because in this case we have Remark 2. In the case q = 1, the solution of the impulsive ordinary differential equation on each interval of continuity could be considered as a solution of the same differential equation with a new initial condition, defined by the impulsive function.It allows the application of induction w.r.t. the interval of continuity.This is different for fractional differential equation.If A1 for IFrDDE is applied, then C t 0 D q x(t) = C t k D q x(t) for t ∈ (t k , t k+1 ) and induction w.r.t. the interval of continuity is not useful.
If A2 for IFrDDE is applied, then induction w.r.t. the interval of continuity could be used.
A detailed explanation of advantages/disadvantages of both the above approaches for equations without delays is given in [19,20].The definition of the solution x(t; t 0 , φ) of the IVP for IFrDDE (1) depends on your point of view.
In this paper, we will use approach A2 for IFrDDE.

Lyapunov Functions and Their Fractional Derivatives
In this paper, we will use piecewise continuous Lyapunov functions [21]): Definition 1.Let α < β ≤ ∞ be given numbers and D ⊂ R n , 0 ∈ D be a given set.We will say that the function V(t, x) : The function V(t, x) is continuous on [α, β)/{t k } × D and it is locally Lipschitz with respect to its second argument; 2.
For each t k ∈ (α, β) and x ∈ D, there exist finite limits In connection with the Caputo fractional derivative, it is necessary to define in an appropriate way the derivative of the Lyapunov functions among the studied equation.The choice is adapted from the case of ordinary differential equations, but it is not applicable since it does not depend on the initial time point (such as the Caputo fractional derivative).
We will give a brief overview of the three main types derivatives of Lyapunov functions V(t, x) ∈ Λ([t 0 − r, T), D) among solutions of fractional differential equations in the literature: ), 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 This type of derivative is applicable for continuously differentiable Lyapunov functions.-second type-Let ψ ∈ C([−τ, 0], D).Then, the Dini fractional derivative of the Lyapunov function where The 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 notation In [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 t 0 , which is typical for the Caputo fractional derivative.The operator defined by ( 9) has no memory.In addition, if x(t) is a solution of (3), then For fractional differential equations without any impulses, notation similar to ( 9) is defined and and ψ ∈ C([−τ, 0], D) be given.Then, the Caputo fractional Dini derivative of the Lyapunov function V(t, x) ∈ Λ([t 0 − r, T), D) is defined by: where or its equivalent The derivative c t k D q (3) V(t, ψ; t 0 , φ 0 (0)) given by ( 11) depends significantly on both the fractional order q and the initial data (t 0 , φ 0 ) of IVP for IFrDDE (3) and it makes this type of derivative close to the idea of the Caputo fractional derivative of a function.

Remark 3.
For any initial data (t 0 , φ 0 ) ∈ R + × C([−τ, 0], D) of the IVP for IFrDDE (3), the relation between the Dini fractional derivative defined by (8) and for any t ∈ J k , ψ ∈ C([−τ, 0], D) and the Caputo fractional Dini derivative defined by ( 11) is given by In the next example, to simplify the calculations and to emphasize the derivatives and their properties, we will consider the scalar case, i.e., n = 1.
Case 3. Caputo fractional Dini derivative.Use (11) and we obtain c

Comparison Results for Delay Fractional Differential Equations
First, we will prove several comparison results for fractional delay differential equation without any impulses.We will use Lyapunov function with the Razumikhin condition Remark 4. A comparison result is given in Theorem 4.5 [22] by applying definition (9) for the derivative of V and incorrectly replacing it with the Caputo derivative in the proof.Some comparison results applying A1 for IFrDDE are obtained in [17], but induction w.r.t. the interval of continuity is incorrectly used (see Remark 2).
Consider the IVP for the following delay fractional differential equation (FrDDE) where Lemma 1. (Comparison result with the Caputo fractional derivative) Assume: 1.
Proof.The proof is similar to that in Lemma 1.The main difference is in connection with inequality (23).Follow the proof in Lemma 1 and in this case we use Lemma 2 and obtain Now, using c It remains to show that we have a contradiction.To see this for any t ∈ [τ 0 , t * ] and h > 0, we let → 0 as h → 0.Then, for any t ∈ [t 0 , t * ], we obtain Since V is locally Lipschitzian in its second argument with a Lipschitz constant L > 0, we obtain Substitute ( 29) in ( 28), divide both sides by h q , take the limit as h → 0 + , use ∑ ∞ r=0 q C r z r = (1 + z) q if |z| ≤ 1, and we obtain for any t ∈ [τ 0 , t * ] the inequality

Comparison Results for Impulsive Delay Fractional Differential Equations
Now, we will prove some comparison result for IFrDDE (3) using approach A2 for IFrDDE.1.

Problem Formulation
We will study neural networks modeled by impulsive Caputo fractional differential equations with bounded time dependent delays and distributed delays.We will consider the case when the lower limit of the fractional derivative is changed after each impulse, i.e., we will use approach A2 for IFrDDE.Following the notations in (3), we consider the general model of Hopfield's graded response neural networks with impulses and bounded delays and distributed delays (INND) where n represents the number of neurons in the network, x i (t) is the pseudostate variable denoting the average membrane potential of the i-th neuron at time t, x(t) = (x 1 (t), x 2 (t), . . ., x n (t)) ∈ R n , q ∈ (0, 1), c i (t) > 0, i = 1, 2, . . ., n, is the self-regulating parameter of the i-th unit, they correspond to the rate with which the i-th neuron rests its potential in the resting state in isolation, a ij (t), b ij (t), i, j = 1, 2, . . ., n, correspond to the synaptic connection strength of the i-th neuron to the j-th neuron at time t and t − τ j (t), respectively, f j (x), g j (x), h j (x) are nonlinear activation functions such that f (x) = ( f 1 (x 1 ), f 2 (x 2 ), . . ., f n (x n )), h(x) = (h 1 (x 1 ), h 2 (x 2 ), . . ., h n (x n )), g(x) = (g 1 (x 1 ), g 2 (x 2 ), . . ., g n (x n )); I = (I 1 , I 2 , . . ., I n ) is an external bias vector, τ j (t) represents the transmission delay along the axis of the j-th unit and satisfies 0 ≤ τ j (t) ≤ r, the t k , k = 1, 2, . . ., are points of acting the state displacements, the functions Φ k,i (t, u, v), k = 1, 2, . . .are the impulsive functions giving the impulsive perturbation of the i-th neuron on the point t k , the numbers x i (t k − 0) = x i (t k ) and x i (t k + 0) are the state of the i-th neuron before and after impulsive perturbation at time t k ; K ij (.) is the delay kernel with . ., n are the initial functions.

Mittag-Leffler Synchronization
Definition 2. The master impulsive Caputo fractional system (40) and the slave impulsive Caputo fractional system (41) are globally Mittag-Leffler synchronized if for any initial functions Remark 8.The synchronization of the problem (40) is studied in [11] and the authors consider the case of constant strengths between the neurons and linear impulsive functions.They used approach A1 for IFrDDE.The main result is based on incorrectly citing and using the Lemma from [17] where they use the derivative (9), which is different than the Caputo fractional derivative (see Remarks 2 and 4).
The main goal of the paper is to implement appropriate controllers u i (t), i = 1, 2, . . ., n for the response system, such that the controlled response system (41) could be synchronized with the drive system (40).

Output Coupling Controller
Inspired by the ideas in [24], the control inputs in the response system are taken as output coupling u j (t) = ∑ n j=1 m ij ( f j (y j (t)) − f j (x j (t))), i = 1, 2, . . ., n.The synchronization via output coupling is important because, in many real systems, only output signals can be measured.
The case of multiple time constant delays (no distributed delays) and the constant synaptic connection strength between neurons is studied in [22] by using quadratic Lyapunov function.We will study the case of variable bounded synaptic connection strength and nonlinear impulsive functions.Theorem 1.Let assumptions A1-A5 be satisfied.
Then, the master impulsive Caputo fractional system (40) and the slave impulsive Caputo fractional system (41) are globally Mittag-Leffler synchronized.( For any natural number k and x ∈ R n , x = (x 1 , x 2 , . . ., x n ) according to condition A4 and Remark 9, we have V(t k , x + L k (x)) ≤ ∑ n i=1 A 2 k,i x 2 i ≤ V(t, x).According to Lemma 5, the claim of Theorem 2 follows.

Lemma 4 .
(Comparison result with the Caputo fractional derivative) Assume: