Improved synchronization criteria for fractional-order complex-valued neural networks via partial control

In this article, without dividing a complex-valued neural network into two real-valued subsystems, the global synchronization of fractional-order complex-valued neural networks (FOCVNNs) is investigated by the Lyapunov direct method rather than the real decomposition method. It is worth mentioning that the partial adaptive control and partial linear feedback control schemes are introduced, by constructing suitable Lyapunov functions, some improved synchronization criteria are derived with the help of fractional differential inequalities and L’Hospital rule as well as some complex analysis techniques. Finally, simulation results are given to demonstrate the validity and feasibility of our theoretical analysis.


Introduction
During the past few years, real-valued neural networks (RVNNs) have attracted much attention due to the background of a wide range of applications such as associative memory, pattern recognition, image processing and model identification [1][2][3][4][5][6]. Since Pecora and Carroll introduced a method to realize synchronization of two identical chaotic systems with different initial conditions [7], synchronization has become a widely studied topic. Up to now, there are many sorts of synchronization, such as complete synchronization [8], projective synchronization [9], and quasi-synchronization [10]. Correspondingly, several different approaches have been used for achieving synchronization, for example, linear feedback control [11], partial control [12] and adaptive control [13]. It is well known that the advantage of adaptive control is that the control parameters can adjust themselves according to the updating laws, which are designed according to the characteristics of the considered system. As we know, many applications of neural networks refer to complexvalued signals, which cannot be handled well by RVNNs, complex-valued neural networks (CVNNs) are such neural networks that their states, connection weights and activity functions are all complex-valued [14,15]. Compared with RVNNs, CVNNs have more complicated properties, and can handle complex signals better. In recent years, the study on CVNNs has attracted some attention, and some important and interesting results have been obtained [16][17][18]. In [18], Song et al. studied the global exponential stability for the addressed CVNNs based on Lyapunov direct method.
Compared with integer-order derivative, fractional-order derivatives can better depict processes and materials possessing hereditary and memory characteristics [19][20][21], these characteristics make fractional-order systems promising candidates for describing some real-world phenomena [22,23]. Some recent studies regarding synchronization control and optimal control for fractional-order systems may be found in [24][25][26]. Lately, many researchers consider fractional-order complex-valued neural networks (FOCVNNs), and some remarkable results on bifurcation [27] and stability [28] as well as synchronization [29][30][31] have been reported. We note that the key technique in [27][28][29][30][31] is to transform the considered FOCVNNs into two equivalent real-valued subsystems, and then their dynamics are studied by employing methods of dealing with fractional-order real-valued neural networks (FORVNNs), that is, the real decomposition method. In [32], Li et al. designed a linear feedback controller and adaptive controller, and the complete synchronization and quasi-projective synchronization criteria for FOCVNNs were derived by using the Lyapunov direct method, respectively.
To the best of our knowledge, a neural network may have a large amount of neurons, it is impossible and unnecessary to impose controllers on all neurons in such large-scale network. To reduce the number of controlled neurons, partial control should be considered. Till now, partial control for neural networks remains little investigated [33,34]. In [34], Wu et al. discussed the stability control for FORVNNs by employing partial linear feedback control strategy. However, to the best of the authors' knowledge, there is no result concerning synchronization problem for FOCVNNs by using partial control scheme. Motivated by the above discussions, this paper investigates the global synchronization for the addressed FOCVNNs by partial control. The main contributions of this paper are the following three aspects: (1) The partial adaptive control and partial linear feedback control schemes are first proposed to realize synchronization of FOCVNNs. (2) FOCVNNs are investigated by Lyapunov direct method rather than real decomposition method, which can reduce computational complexity. (3) Several succinct synchronization criteria for FOCVNNs are derived.
The organization of the paper is as follows. In Sect. 2, the model formulation and preliminary results are presented. In Sect. 3, the partial adaptive control and partial linear feedback control schemes are proposed to achieve synchronization for the addressed FOCVNNs. In Sect. 4, numerical simulations are provided to illustrate the effectiveness of our theoretical results. Finally, conclusions are given in Sect. 5.

Preliminaries and model description
In this section, some definitions and lemmas are recalled, which will be needed later.
Definition 1 ([35, 36]) The Riemann-Liouville fractional integral of the function w(t) is defined by

Definition 2 ([35, 36]) The Caputo fractional derivative of function w(t) is defined by
Next, we consider a FOCVNN described by or in a compact form . . , f n (w n (t))) T : C n → C n denotes the vector-valued activation function, and the activation f p (·) satisfies the following assumption.
Assumption 1 For any w, u ∈ C, there exists a positive constant l p satisfying For simplicity, we refer to FOCVNN (2) as the drive system, and the controlled response system is given as or, in a compact form, where z(t) = (z 1 (t), z 2 (t), . . . , z n (t)) T ∈ C n is the state vector, u(t) = (u 1 (t), u 2 (t), . . . , u n (t)) T ∈ C n is the control input vector. Now, we introduce some lemmas, which will be used in the proof of our main results.

Lemma 5 ([39]) For a symmetric matrix W ∈ R n×n and a diagonal matrix K
where W l is the minor matrix of W by removing its first l row-column pairs, B and G are matrices with where 0 < q < 1, ζ ∈ R and t 0 is the initial time. Then

Main results
In this section, the partial adaptive control and partial linear control strategies are employed to reduce the control costs, and some novel criteria are derived to ensure the global synchronization of FOCVNNs (2) and (4).
Define the error e j (t) = u j (t)w j (t) for j ∈ Λ, and design the partial adaptive controller u j (t) as follows: where η j > 0, ξ j (t) ∈ R.
Theorem 1 Under Assumption 1 and the partial adaptive controller (5), FOCVNNs (2) and (4) can achieve the global synchronization if where (AA H ) l is the minor matrix of matrix AA H by removing its first l (1 ≤ l < n) rowcolumn pairs.

Corollary 1 Under Assumption
is satisfied, then FOCVNNs (2) and (4) can achieve the global Mittag-Leffler synchronization under the partial linear feedback controller (20).
Proof Construct a Lyapunov function in the following form: Using a similar process with respect to (9)-(12), we derive ). According to Lemma 6, we have V (t) ≤ V (t 0 )E q (-λ 1 (tt 0 ) q ), the proof of Corollary 1 is completed.
Remark 3 We observe that the existing control schemes for neural networks almost control all neurons, we realize synchronization for the addressed FOCVNNs in this paper by employing partial linear control and partial adaptive control schemes.
If we take l = n, that is, all neurons are controlled, then the partial adaptive controller (5) becomes the adaptive controller where η j > 0, ξ j (t) ∈ R. In this case, we can derive the following corollary. proof with respect to (13)- (19), we derive lim t→+∞ e H (t)e(t) = 0, the proof of Corollary 2 is completed.
Remark 5 In [23,[29][30][31], the authors obtained synchronization criteria by real decomposition method. Compared with the real decomposition method in [23,[29][30][31], the Lyapunov direct method employed in this paper is more nature and compact. In [23], the authors adopted adaptive feedback control strategy, and realized synchronization for the considered FOCVNNs. In [32], the authors employed linear feedback control strategy, and achieved synchronization for the considered FOCVNNs. Compared with control strategies in [23,32], our partial adaptive control strategy is more easy and has less cost from the point of view of practical applications.

Conclusion
This paper is concerned with the global synchronization of FOCVNNs. To realize the synchronization goal, the partial adaptive controller and partial linear feedback controller are designed, respectively. On the basis of the Lyapunov method, the L'Hospital rule and some complex analysis techniques, some succinct criteria are derived to ensure the global synchronization for the considered FOCVNNs. Numerical simulations are given to show the effectiveness and feasibility of the proposed method. The topic concerning dynamics analysis for FOCVNNs is of importance, we will make some efforts on the finite-time synchronization of FOCVNNs.