Adaptive synchronization of fractional-order complex-valued neural networks with time-varying delays

In this paper, the adaptive synchronization of fractional-order complex-valued neural networks with time-varying delays(FOCVNNTDs) is investigated. First, two novel fractional-order differential inequalities with time delays are established, which can be seen as an extension of Halanay inequality. Besides, complete synchronization and quasi-projective synchronization of FOCVNNTDs are investigated based on the two novel inequalities using a novel adaptive controller. In addition, instead of separating the complex-valued neural networks into two real-valued networks, a non-decomposition method is adopted to study the adaptive synchronization of FOCVNNTDs, which avoids the difficulty and complexity of theoretical analysis. Finally, some numerical simulation examples are provided to demonstrate the validity of our theory.


I. INTRODUCTION
A RTIFICIAL neural networks have always been a research hot spot in artificial intelligence due to their properties of simulating biological neural networks. They have been widely applied in pattern recognition [1], signal processing [2] and other hot fields. However, time delays are inevitable in both biological neural networks and their hardware implementations. Additionally, time-varying delays may cause chaotic, oscillating, or unstable behaviors of the system. Thus, it is necessary and meaningful to take time delays into consideration when studying neural networks. And numerous papers involving neural networks with time delays have been published [3]- [7].
Fractional-order calculus was initially proposed as a purely mathematical theory, but scholars are constantly exploring the research value of fractional systems in practical applications over the years. So far, fractional calculus has been extensively utilized in fluid mechanics [8], finance [9], infectious diseases [10], [11] and other fields. In contrast to neural network models with integer-order, fractional-order models can more accurately describe the change process of the memory and historical characteristics of the natural system. Hence, it is of great significance to investigate fractionalorder neural networks(FONNs).
Recently, the dynamic behaviors of FONNs such as dissipativity [12], synchronization [13]- [15] and stability [16], [17] have attracted increasing interest. Among all these dynamic behaviors, synchronization refers to a certain relative relationship between two or more time-varying systems under external control. Based on this property, synchronization is widely applied in image processing [18], security communications [19] and other fields. Currently, various synchronizations have been investigated such as exponential synchronization [20], [21], Mittag-Leffler synchronization [22]- [24], complete synchronization [25] and projective synchronization [26]. In [27]- [30], the finite-time synchronization of FONNs was studied. Moreover, in [31], [32], the authors attempt to investigate the fixed-time synchronization of FONNs via sliding mode control and integer-order methods. Among these synchronizations, the projective synchroniza-tion is of essential research and application value due to its characteristic, which is that the drive and response systems can achieve proportional synchronization more quickly.
To synchronize systems, numerous control methods have been proposed, including feedback control [29], [33], impulse control [34], adaptive control [24] and sliding mode control [32]. Above all these strategies, the adaptive control is an excellent choice to study the synchronization of FONNs. Because it can update the control gains by itself and is easier to be applied in the existing systems. Therefore, it is valuable to design a suitable adaptive controller to study synchronization problems of neural networks.
It is worth noting that complex-valued signals are commonly found in pattern recognition [35], nonlinear filtering [36] and image reconstruction [37]. Compared with realvalued systems, complex-valued systems have more complex properties and wider practical applications. When studying complex-valued neural networks, a number of researchers choose to separate the networks into two real-valued systems [13], [38]- [40]. Although it is feasible, the number of dimensions of the system is doubled, which greatly increases the difficulty and complexity of theoretical analysis. Therefore, in [29], [33], [41], the authors use a non-decomposition method, which is based on the proposed complex-valued inequalities, to avoid this problem.
In [33], the author studies the quasi-projective synchronization of fractional-order complex-valued neural networks without accounting for time delays in the model. For neural networks with time delays, researchers usually use the following methods to deal with the delays. One approach is to utilize a controller with a delay term to eliminate the time delays in the networks [24], [42]. However, the controller designed to solve the time delay problem is not simple enough. The second method is to use the inequality scaling technique to eliminate terms with time delays [43]. The third method is to use the existing inequality lemmas to solve the time delay problem. Therefore, in the research of synchronization of neural networks, it is worth studying which method to use to deal with time delays. And it is necessary to apply a simple controller and rigorous mathematical proofs to achieve synchronization of neural networks.
Following the analysis above, this article study the adaptive synchronization of FOCVNNTDs. A summary of the main contributions of this paper is: (1) Firstly, a novel fractional differential inequality with time delays (Lemma 7) is established. Different from the results in [33], [43], this novel inequality contains time delays, which are inevitable in natural systems. Based on this inequality, we can design a controller without time delay, which saves control costs. Moreover, in the proof of this paper, as long as the condition of this inequality is satisfied, the synchronization result can be obtained.
(2) Secondly, another novel fractional differential inequality (Lemma 8) is established. Compared to Lemma 7, this inequality has a more common form. Under certain conditions, Lemma 8 can degenerate into Lemma 7, which extends our theoretical results. In addition, as a generalization of fractional Halanay inequality in [44], these two novel inequalities are suitable for investigating the adaptive synchronization problem of fractional-order systems.
(3) Thirdly, complete synchronization and quasi-projective synchronization of FOCVNNTDs are investigated based on novel adaptive controllers and these two novel fractional differential inequalities. Compared to the results in [24], [43], the controllers we designed are more straightforward and the theoretical proof process in the paper is rigorous. This paper will be organized in the following structure. Section 2 of presents some definitions, known lemmas and two new fractional differential inequalities. In Section 3, two adaptive controllers are designed and some adaptive synchronization criteria of FOCVNNTDs are established. Section 4 will provide some numerical simulation examples to demonstrate the validity of our theory. In Section 5, we reach a conclusion.
Notation: In this article, the sets C and C n , respectively, denote all complex numbers and a space containing all ndimensional complex vectors, while R and R + denote the set of all real numbers and the set of all real numbers that are non-negative. For z ∈ C, the real part of z is Re(z) whereas the imaginary part is Im(z),z is the conjugate of z, |z| = √ zz represents the modulus of z.

A. PRELIMINARIES
A few basic definitions and necessary lemmas will be presented in this part.

Definition 2.
[45], [46] When α ∈ (0, 1), the Caputo fractional derivative with fractional-order α for a differentiable function and the Laplace transform of C 0 D α t f (t) is given as where f (s) is the Laplace transform of f (t).

Lemma 4. [33]
The function f (t) ∈ C is analytic and continuous, then the following inequality is established: where the function ψ(t) is continuous and bounded, σ is a positive constant. The coefficients satisfy Moreover, if lim t→+∞ (t − τ (t)) = +∞, then for any given ε > 0, there exists t * = t * (M 0 , ε) > t 0 such that In addition, the inequality still holds if the conditions is modified into that Proof. According to (1), there exists a non-negative function Based on the inverse Laplace transform in Lemma 1, we can get where * is convolution operator. Based on the properties of Mittag-Leffler function, we know that for t ≥ t 1 and t 1 = Γ(α) Obviously, lim t→+∞ a (t) = 0, K (t) ≥ 0 and lim t→+∞ K (t) = 0. According to Lemma 5, we need to prove the condition On the other hand, from Definition 3, We get According to (5), (7) and ρ > µ, From Lemma 5, lim t→+∞ V (t) = 0, the proof is completed.
is a non-negative function and ε can be selected as a positive number that is infinitely close to zero, we get lim t→+∞ V (t) = 0. That means Lemma 8 can be reduced to Lemma 7.

B. MODEL DESCRIPTION
In this article, the following FOCVNNTDs is considered: for t ≥ 0, where 0 < α < 1, j ∈ {1, 2, ..., n}, n represents the number of neurons, f j (·) , g j (·) : C → C respectively denote the activation functions without and with delays, x i (t) ∈ C denotes the state of the ith neuron at time t , d i ∈ R denotes the self-inhibition rate of the ith neuron, a ij , b ij ∈ C represent the connection weight of the ith neuron and jth neuron, τ j (t) is time-varying delay satisfying 0 ≤ τ j (t) ≤ τ , I i (t) ∈ C denotes the external input.
Let system (16) be the drive system, and the response system is described as: a ij f j (y j (t)) + I i (t) where y i (t) ∈ C denotes the state variable of the ith neuron of the system (17) at time t, and u i (t) ∈ C denotes the controller.
To study the synchronization between system (16) and system (17), some assumptions are needed: Assumption 1. For any i, j ∈ {1, 2, ..., n}, u ∈ C, there exist real numbers m 1j , m 2j > 0 such that For the external input I i (t), we have where γ i is a real number.

Assumption 2.
The activation functions f j (·) and g j (·) are Lipschitz continuous, and for any u, v ∈ C, there exist real numbers l 1j , l 2j > 0 such that

III. MAIN RESULTS
In this part, based on the new lemmas, the synchronization problem of FOCVNNTDs is investigated by adaptive controller.

A. THE COMPLETE SYNCHRONIZATION OF FOCVNNTDS BY ADAPTIVE CONTROL
The synchronization error is e i (t) = y i (t) − x i (t). Design the adaptive controller as: where m i (t) is the adaptive coefficient and k i is a positive constant. Based on the above conditions, we get that Proof. Construct the Lyapunov function as: where m 1 satisfies the inequality (23).
ing to Lemma 2 and Lemma 3, the α-order Caputo derivative According to Lemma 2 and Assumption 2, then we can get the inequality (21) . That is and µ = n max 1≤i≤n l 2i 2 . We choose appropriate parameters to meet then according to Lemma 7, we have Therefore, we can obtain that lim t→+∞ e(t) 2 = 0.
Based on the above analysis, it is clear that system (16) and (17) are completely synchronized. The proof is completed.

B. QUASI-PROJECTIVE SYNCHRONIZATION OF FOCVNNTDS BY ADAPTIVE CONTROL
This part investigates the quasi-projective synchronization of FOCVNNTDs by adaptive control. Define the synchronization error asê where β ∈ C is the projection coefficient and β = 0. The adaptive controller is: where m i (t) is adaptive coefficient and k i is a positive constant. According to these conditions, we get a ij [f j (βx j (t)) − βf j (x j (t))] (26) For convenience, we define: µ = n max 1≤i≤n l 2i 2 , Theorem 2. According to Assumption 1 and Assumption 2, system (16) and (17) can achieve quasi-projectively synchronized under the adaptive controller (25) if ρ 2 > µ ≥ 0 and C ≥ 0. In addition, we can obtain that where the number ε is positive.
Proof. Construct the Lyapunov function as: where m 2 satisfies the inequality (35).
ing to Lemma 2 and Lemma 3, the α-order Caputo derivative From Assumption 2 and applying Lemma 2, (30) According to Assumption 1 and applying Lemma 2, we derive and (32) According to Lemma 2, we can get Submitting (29)-(33) into (28), it has Then, choose m 2 such that and C ≥ 0 . According to Lemma 8, there exists t 2 = Γ(α) ρ2 Moreover, if lim t→+∞ (t − τ (t)) = +∞, then for any given Thus, based on (27), Based on the property of Mittag-Leffler function, we can finally get where ε is a positive number. Therefore, system (16) and system (17) can achieve quasi-projective synchronization with the error bound C ρ2−µ + ε. The proof process is finished.
Remark 4. Instead of separating the complex-valued networks into two real-valued systems [38]- [40], the proofs of Theorem 1 and Theorem 2 apply a non-decomposing method based on complex functions to deal with complex-valued systems, which simplifies the proof process and reduces the difficulty of theoretical analysis.
Remark 5. In [24], the author designs an adaptive controller with time delays to study the Mittag-Leffler synchronization of delayed FONNs. In contrast, the controllers used in Theorem 1 and 2 do not contain time delay terms. This means that our controller is simpler and can save control costs to some extent. Remark 6. On the processing of time delays in [43], the author gives ξ > 1 such that V 1 (t − τ ) ≤ ξV 1 (t). In this way, the items with time delays in the proof are eliminated. This seems feasible, but is not rigorous enough because we do not know how to determine the value of ξ. Based on the conclusions of Lemma 7 and 8, we do not need to eliminate the time delay terms during the proofs of Theorem 1 and 2. Therefore, Theorem 1 and 2 are more rigorous and reasonable in handling time delays due to the two novel inequalities.
The phase trajectories of real part Re(x i (t)) and imaginary part Im(x i (t))(i = 1,2) of system (40) are shown in Figs.1 and 2.