Finite-Time Mittag–Lefﬂer Synchronization of Neutral-Type Fractional-Order Neural Networks with Leakage Delay and Time-Varying Delays

: This paper studies fractional-order neural networks with neutral-type delay, leakage delay, and time-varying delays. A sufﬁcient condition which ensures the ﬁnite-time synchronization of these networks based on a state feedback control scheme is deduced using the generalized Gronwall–Bellman inequality. Then, a different state feedback control scheme is employed to realize the ﬁnite-time Mittag–Lefﬂer synchronization of these networks by using the fractional-order extension of the Lyapunov direct method for Mittag–Lefﬂer stability. Two numerical examples illustrate the feasibility and the effectiveness of the deduced sufﬁcient criteria.


Introduction
Fractional calculus studies the different possibilities of defining real or complex orders for the differentiation and integration operators. Although it has a long history, only recently it has been successfully applied to physics and engineering problems. As such, in the past few years, it became clear for engineers and scientists that some phenomena can be more accurately described by employing the fractional derivative. Fractional differential equations have been proved to better describe many systems in interdisciplinary fields, such as chemistry, biology, physics, mechanics, electromagnetism, heat transfer, acoustics, economy, and finance.
The finite-time stability and synchronization properties of fractional-order neural networks were intensely studied over the last few years. Concretely, "finite-time stability analysis of fractional-order complex-valued memristor-based neural networks with time delays" was done in [17]. Finite-time stability criteria for fractional-order delayed neural networks were also established in [18][19][20]. A more general model, namely fractional-order Cohen-Grossberg BAM neural networks with time delays was researched in [21], from the finite-time stability point of view. Finite-time stability analysis hand, in neutral-type systems, past derivative information has also been observed to influence the present state. The properties of neural reaction processes that occur in the real world can be more accurately described by these systems. The study of these systems is more complicated than that of the usual time-delayed models because of the existence of the neutral-type delay. This type of delay is relevant in many application domains, like automatic control, population dynamics, and vibrating masses attached to an elastic bar. Neutral delay may also appear when implementing neural networks in VLSI circuits. These facts compelled us to also add neutral-type delay to our model. Taking all the above into account, we consider neutral-type fractional-order neural networks with leakage delay and time-varying delays in this paper, and study their finite-time synchronization and finite-time Mittag-Leffler synchronization based, respectively, on two general state feedback control schemes.
The main contributions of the paper are: (1) the introduction, for the first time in the literature, to the best of our knowledge, of the fractional-order neural networks with neutral-type delay, leakage delay, and time-varying delays; (2) the use of the generalized Gronwall-Bellman inequality to deduce sufficient criteria for the finite-time synchronization of the introduced networks, using a general state feedback control scheme; (3) the use of the fractional-order extension of the Lyapunov direct method for Mittag-Leffler stability in order to deduce sufficient criteria for the finite-time Mittag-Leffler synchronization of the introduced networks, using a different general state feedback control scheme; (4) the possible use of the methods developed in this paper for more general models, with impulsive effects, reaction-diffusion terms, or Markovian jump parameters.
The summary of the rest of the paper is the following. The neutral-type fractional-order neural networks with leakage delay and time-varying delays are introduced in Section 2, together with definitions regarding fractional calculus, definitions of the finite-time synchronization and the finite-time Mittag-Leffler synchronization, one assumption about the activation functions, and four useful lemmas. Then, Section 3 is dedicated to the deduction of the sufficient criteria which ensure finite-time synchronization and finite-time Mittag-Leffler synchronization, respectively, of the introduced model. Section 4 details the two numerical examples given to illustrate the feasibility and the effectiveness of the deduced sufficient criteria. The conclusions of the paper are presented in Section 5.
Notations: R denotes the set of real numbers and R n denotes the Euclidean space of dimension n. A T represents the transpose of matrix A. I n denotes the identity matrix of order n and 0 n the empty matrix of order n. That matrix A is positive definite (negative definite) is denoted by A > 0 (A < 0). The smallest eigenvalue of positive definite matrix P is λ min (P). || · || represents the vector Euclidean norm or the matrix Frobenius norm, and | · | is the element-wise vector norm or the element-wise matrix norm.

Preliminaries
First, we will give a few definitions involving fractional calculus.

Definition 1 ([45]
). "The fractional integral of order α for an integrable function x : [t 0 , ∞) → R is defined as: where t ≥ t 0 , α > 0, and Γ(·) is the gamma function, defined by: for Re(τ) > 0, where Re(·) represents the real part." Definition 2 ([45]). "The fractional Caputo derivative of order α for a function x ∈ C n ([t 0 , ∞), R) is defined by: where t > t 0 and n is a positive integer, with n − 1 < α < n. Moreover, when 0 < α < 1, we have that: Definition 3 ([45]). "The Mittag-Leffler function is defined by: where α > 0 and z ∈ C. When α = 1, we have that E 1 (z) = e z ." Now, the neutral-type fractional-order neural networks with leakage delay and time-varying delays will be considered as the master system: for ∀i = 1, . . . , N, where x i (t) ∈ R represents the state of the ith neuron at time t, c i ∈ R + represents the self-feedback weight of the ith neuron, a ij ∈ R is the weight without time delay between the ith and jth neurons, b ij ∈ R is the weight with time delay between the ith and jth neurons, g i ∈ R is the neutral-type weight of the ith neuron, f j : R → R represent the nonlinear activation functions, ∀j = 1, . . . , N, I i ∈ R is the external input for the ith neuron, µ is the leakage delay, τ : R + → R + are the time-varying delays, and η is the neutral-type delay.
In the following, we assume that the time-varying delays τ : R + → R + are continuously differentiable functions and there exist τ > 0 and τ < 1, such that The initial conditions of system (1) are given by

The slave system is given by
for ∀i = 1, . . . , N, and y i (t) ∈ R represents the state of the ith neuron at time t, and u i (t) represents a control input. The initial conditions of system (2) are given by If we denote by e i (t) = y i (t) − x i (t) for ∀i = 1, . . . , N, then, based on the expressions of the master system (1) and the slave system (2), the error system has the form The initial conditions of error system (3) are given by The state feedback control scheme will be used to obtain finite-time synchronization between master system (1) and slave system (2). In this case, the controller is given by where for ∀i = 1, . . . , N. System (5) can be written in matrix form as Definition 4. The master system (1) is said to be finite-time synchronized with the slave system (2) based on the controller (4), if there exist positive constants {δ, ε, T}, 0 < δ < ε, such that ||σ|| < δ implies ||e(t)|| < ε, ∀t ∈ [0, T).
We will also use a different state feedback control scheme to realize finite-time Mittag-Leffler synchronization between master system (1) and slave system (2), for which the controller is given by (7) where k i1 , k i2 , k i3 , k i4 ∈ R + represent the control gains, for ∀i = 1, . . . , N. System (3) becomes in this case for ∀i = 1, . . . , N.
System (9) can be written in matrix form as Definition 5. The master system (1) is said to be finite-time Mittag-Leffler synchronized with the slave system (2) based on the controller (7), if there exist positive constants {δ, ε, λ, β, The following assumption has to be made in order to study the synchronization of the networks defined above. Assumption 1. The following Lipschitz conditions are satisfied by the activation functions f j for any x, x ∈ R: where, ∀j = 1, . . . , N, l j > 0 represent the Lipschitz constants. Also, let L := diag(l 1 , l 2 , . . . , l N ).
We will also need the following lemmas: where t > t 0 and n is a positive integer, with n − 1 < α < n. Moreover, when 0 < α < 1, we have that: 46]). "Suppose α > 0, a(t) is a nonnegative function locally integrable on 0 ≤ t < T (for some T ≤ +∞) and g(t) is a nonnegative, nondecreasing continuous function defined on 0 ≤ t < T, g(t) ≤ M (M is a constant), and suppose u(t) is nonnegative and locally integrable on 0 ≤ t < T with Moreover, if a(t) is a nondecreasing function on 0 ≤ t < T, then where 0 < α < 1." where 0 < α < 1 and λ > 0. Then

Main Results
In the following, we will assume that 0 < α < 1. First, we give a sufficient condition that ensures the finite-time synchronization of master system (1) and slave system (2), based on the controller (4). Theorem 1. If Assumption 1 holds, then master system (1) and slave system (2) are finite-time synchronized based on the controller (4) if ||G − K 4 || < 1 and there exist positive constants {δ, ε, T} such that the following inequality holds: Proof. Integrating relation (6), we get that or, by using Lemma 1, that We have that With the change of variable s − η − u = v, we get Then, with the change of variable v = (t − η − u)w, we get where B(x, y) denotes the Euler beta function. Therefore: By taking the norm of relation (12), and also taking into account the above relation, we obtain that where, for the last inequality, we used Assumption 1.

Remark 1.
The condition in Theorem 1 only needs the computation of the norm of 8 matrices, which is easily done using the default function in MATLAB. Also, the verification of the inequality only needs computing the Mittag-Leffler function for the biggest argument, because it is a strictly increasing function.
We now give a sufficient condition that ensures the finite-time Mittag-Leffler synchronization of master system (1) and slave system (2), based on the controller (7). Theorem 2. If Assumption 1 holds, then master system (1) and slave system (2) are finite-time Mittag-Leffler synchronized based on the controller (7) if and there exist positive constants {δ, , T} such that where λ = 2λ min (K 1 − |A|L).
Proof. The following Lyapunov function will be considered:

t)e(t).
Taking into account Lemma 3, and computing the fractional-order derivative of V(t) along the trajectories of system (9), it follows that where, for the last inequality, we used Assumption 1. Now, taking into account conditions (13), inequality (16) becomes: where λ = 2λ min (K 1 − |A|L) > 0. By Lemma 4, we have that or, equivalently, which further leads to where, for the last inequality, we used condition (14). This completes the proof of the theorem.

Remark 2.
The condition in Theorem 2 only needs to verify that 4 matrices are positive definite. Again, the verification of the inequality only needs computing the Mittag-Leffler function for the biggest argument, because it is a strictly increasing function.

Numerical Examples
Two numerical examples shall be given in this section to illustrate the feasibility and the effectiveness of the sufficient criteria deduced above. Example 1. The two-neuron neutral-type fractional-order neural network having leakage delay and time-varying delays will be the master system: the fractional-order neural network with two neurons will be the slave system: and the controller will be: where e(t) = y(t) − x(t), and α = 0.5, from which we deduce that L = 0.25 0 0 0.25 , and so Assumption 1 is fulfilled. The leakage delay is µ = 0.04, the time-varying delays are τ(t) = 0.1| cos t|, and the neutral-type delay is η = 0.05, from where we get that τ = τ = 0.1 and ω = max{µ, τ, η} = 0.1. The values of K 1 , K 2 , K 3 , K 4 are designed as: With these values we get that ||G − K 4 || = 0.6 < 1, and, if we take δ = 0.1 and ε = 100, then condition (10) holds for T = 2.1411, which means that the conditions of Theorem 1 are fulfilled. Thus, we obtain that master system (16) is finite-time synchronized with slave system (17), based on controller (18).
The state trajectories and the phase trajectories of the errors e 1 and e 2 are depicted in Figures 4-6, for 8 initial values.

Conclusions
Two sufficient criteria which ensure the finite-time synchronization and finite-time Mittag-Leffler synchronization of fractional-order neural networks with neutral-type delay, leakage delay, and time-varying delays were given, by making the assumption that the activation functions satisfy the Lipschitz conditions. The generalized Gronwall-Bellman inequality was used to realize the finite-time synchronization of the introduced networks, based on a general state feedback control scheme. Then, the fractional-order extension of the Lyapunov direct method for Mittag-Leffler stability was used to realize the finite-time Mittag-Leffler synchronization of the same networks, based on a different general state feedback control scheme. The feasibility and the effectiveness of the theoretical results was illustrated by providing two numerical examples.
The methods developed in the paper are general, and can be used to obtain sufficient criteria for the finite-time synchronization and the finite-time Mittag-Leffler synchronization of neural network models with impulsive effects, reaction-diffusion terms, or Markovian jump parameters. These developments represent promising future work directions.