Finite-Time Synchronization for Complex-Valued Recurrent Neural Networks with Time Delays

This paper focuses on the finite-time synchronization analysis for complex-valued recurrent neural networks with time delays. First, two kinds of common activation functions appearing in the existing references are combined together and more general assumptions are given. To achieve our aim, a nonlinear delayed controller with two independent parameters different from the existing ones is provided, which leads to great difficulty. To overcome it, a newly developed inequality is used. Then, via Lyapunov function approach, some criteria are derived to guarantee the finite-time synchronization of the considered system, and the settling time for synchronization is also estimated. Finally, two numerical simulations are given to support the effectiveness and advantages of the obtained results.


Introduction
In recent years, neural networks have been always very important to researchers in various fields, to name a few, control theory and signal processing in electrical engineering, parallel computation and pattern recognition in computer science, and modeling optimization problems in applied mathematics.Moreover, as everyone knows, most applications rely heavily on the dynamical behavior of recurrent neural networks.This sets off a research upsurge for the dynamics of neural networks.As a result, many researchers have put their efforts into the analysis and synthesis problems for the dynamics of neural networks for several decades; see [1][2][3][4][5][6][7][8] and the references therein.
In most results considering the stability and the stabilization of neural networks, the convergent mode is asymptotically stable and the time for system trajectories to reach an equilibrium point is infinite.However, in practical problems, the convergent time is often required to be faster or even finite.Thus, so as to satisfy this requirement, the topic about finite-time stability appeared.Recently, finite-time stability and synthesis problems for various systems have been hot issues and attracted many scholars' interests .Among them, finite-time stability, stabilization, synchronization and robust passive control for time-delay systems, stochastic systems, chaotic systems, multiagent systems, and nonlinear systems were studied in [10][11][12][13][14][15][16][17][18][19][20].For various neural networks models, finite-time stabilization, instabilizability, adaptive control,  ∞ control, and state estimation were discussed in [22][23][24][25][26] and finite-time synchronization control problems were addressed in [27][28][29].
On the other hand, a complex-valued recurrent neural networks model has produced a research upsurge for the past few years.The main feature is that it owns complexvalued states, connection weights, and activation functions.This decides that this kind of system can be applied to a wider range involving electromagnetic waves, radar imaging, quantum waves, and so on, but it also leads to more difficulties in analyzing its dynamical behaviors in contrast to real-valued systems.In a word, anyway, the occurrence of complexvalued neural networks draws a great deal of attention from all aspects.Accordingly, many fruitful achievements have been reported for this model .For example, global stability, global exponential stability, and Hopf bifurcation problems for complex-valued neural networks systems with delays or without delays were investigated in [32][33][34][35][36][37][38][39][40].Dissipativity, Complexity passivity, state estimation, exponential stability, and inputto-state stability for memristor-based complex-valued neural networks with or without delays were discussed and relative criteria were established in [43][44][45][46][47][48].However, for finite-time stability analysis of delayed complex-valued neural networks models, until now, few results have been developed.The boundedness problem of these systems in finite-time interval is considered in [49,50].The finite-time synchronization problem for complex-valued neural networks (CVNNs) with infinite-time distributed delays is discussed in [51] and the finite-time stabilizability and instabilizability for complexvalued memristive neural networks with time delays are studied in [52].Moreover, considering the rich background of complex-valued neural networks models and the present status of development for them, it is absolutely essential to explore the dynamics of these models deeply.Thus, these further inspire us to make deeper research.
In this article, in view of the unavoidability of time delays [53][54][55], the finite-time synchronization problem for complex-valued recurrent neural networks with time delays is presented.The main contributions of our work are embodied in several points: (1) A nonlinear delayed controller with independent parameters  1 and  2 different from the most existing results is designed to achieve the finite-time synchronization of the considered system.(2) To deal with the difficulty involved by the contribution (1), a new inequality stated in Lemma 6 proposed and proved by us in [52] is used.
(3) For real-imaginary separate-type activation functions, two kinds of common functions in the existing references are combined together and more general conditions are assumed in this paper.So the proposed results have broader applications.
The rest of this article is organized as follows.Section 2 provides some preliminaries.Section 3 presents main criteria about the finite-time synchronization for delayed complexvalued recurrent neural networks and estimates the settling time by a new controller.Section 4 gives two numerical simulations to show the validity of theoretical results.Section 5 concludes this paper.

Preliminaries
Consider a complex-valued recurrent neural networks model as the drive system represented by with the initial conditions where  =      . ( Remark .For the dynamic analysis of complex-valued recurrent neural networks models, the most results are based on the existence, continuity, and boundedness of the partial derivatives for real and imaginary parts [32,33,46,49].Then, by the mean value theorem, it can be proved that they satisfy the inequalities similar to those in Assumption 1.
In fact, only the inequalities can be used in the process of obtaining the main results.Thus, the existence, continuity, and boundedness of the partial derivatives are unnecessary and they also lead to limitations in choosing complex-valued activation functions.Here, we remove these constraints and provide a more suitable assumption, Assumption 1, made on the activation functions, so that the current work can be applied to solve more engineering problems.This advantage can be seen from example 2.
Remark .It should be noticed that such an assumption is made on the activation functions in the literature [44,45]; i.e., ℎ  () = ℎ   (()) + ℎ   (()),   () =    (()) +    (()) for the complex number  and the following inequalities hold: for all  = 1, 2, . . ., , where Obviously, this assumption is a special case of Assumption 1.Therefore, in this paper, we will not discuss this case in detail, but a numerical example for this case will be provided to show the effectiveness.
Remark .In this paper, we design the controller (15) with independent parameters  1 and  2 , which are different from the most existing results based on  1 =  2 , such as [17,18,22].This is one of the main features for our work, which also leads to difficulty in studying finite-time dynamic behavior of the considered system.To solve this difficulty, we use the inequality in Lemma 6 proposed and proved by us in [52].
First, for convenience, we denote Based on the controller (15), the main result of our paper can be obtained as follows.
Remark .Recently, the finite-time stabilization and synchronization problems of real-valued neural networks have been investigated and the time for system trajectories tending to an equilibrium point is finite [22][23][24][25][26][27][28][29].However, for finite-time stability analysis of complex-valued neural networks models, until now, few results have been developed.For instance, the finite-time synchronization problem for complex-valued neural networks (CVNNs) with infinitetime distributed delays is discussed in [51] and the finitetime stabilizability and instabilizability for complex-valued memristive neural networks with time delays are studied in [52].Here, Theorem 10 further provides sufficient conditions for the finite-time synchronization of the considered system and their effectiveness can be illustrated in numerical examples.From the point of theory development, our result is significant in dynamical behavior analysis of complex-valued models and it can be regarded as the extension of the previous work.Example .Consider complex-valued neural networks as the drive system given by
By choosing the parameters of the controller (15) as  11 = 2.3,  12 = 1.6,  21 = 1.1,  22 = 0.8,  11 =  12 =  21 =  22 = 1,  11 = 1,  12 = 0.6,  21 = 1.5,  22 = 1.6,  1 = 0.5, and  2 = 0.6, it is easy to check that the conditions of Theorem 10 are satisfied.Then, under the corresponding controller, the drive system (28) can be synchronized with the response system (30) in finite time.In addition, according to Theorem 10 and Remark 7, it can be computed that  1 ≤ max{10.88,10.95} = 10.95 seconds.The synchronization curves between the drive system (28) and the response system (30) are displayed in Figure 3.The time responses of the synchronization errors between them are shown in Figure 4. From Figures 3 and 4, it can be seen that system (28) can synchronize with system (30) in finite time via the corresponding controller under the given initial values.These also further show that the obtained result is effective and our work is meaningful.
Next, one example with special activation functions stated as in Remarks 2 and 3 is given to illustrate the validity of our result.
By choosing the parameters of the controller (15) as  11 = 4.2,  12 = 6.5,  21 = 4.4,  22 = 5.9,  11 =  12 =  21 =  22 = 1,  11 = 2.5,  12 = 3.1,  21 = 2,  22 = 3.6,  1 = 0.5,  2 = 0.6, it is easy to check that the conditions of Theorem 10 are satisfied.Thus, under the corresponding controller, the drive system (31) can be synchronized with the response system (33) in finite time.Moreover, according to Theorem 10 and Remark 7, we get  2 ≤ max{10.95,11.19} = 11.19 seconds by simple calculation.The synchronization curves between the drive system (31) and the response system (33) are shown in Figure 7.The time responses of the synchronization errors between them are depicted in Figure 8. From Figures 7  and 8, it can be seen that system (31) can synchronize with system (33) in finite time via the corresponding controller under the given initial values.This example also further illustrates the effectiveness and superiority of our proposed result.

Conclusion
In this paper, the finite-time synchronization problem of complex-valued recurrent neural networks with time delays has been studied.Based on the more general assumptions for activation functions, a nonlinear controller with independent parameters, and a new inequality proposed and proved by us in [52], some sufficient conditions have been established and the settling time for synchronization has been estimated.The obtained results have been shown to be effective and superior by two examples.Moreover, it is well known that the systems with stochastic terms are more extensive in practical applications [56][57][58][59][60], such as the noises, and we will discuss the addressed models with noises deeply in the future.

Figure 3 :
Figure 3: Synchronization curves of real and imaginary parts of  1 ,  2 , z1 , and z2 under controller for example 1.

2 Figure 4 :
Figure 4: Time responses of synchronization errors  1 and  2 under controller for example 1.

Figure 7 :
Figure 7: Synchronization curves of real and imaginary parts of  1 ,  2 , z1 , and z2 under controller for example 2.

Figure 8 :
Figure 8: Time responses of synchronization errors  1 and  2 under controller for example 2.