Finite time synchronization of memristor-based Cohen-Grossberg neural networks with mixed delays

Finite time synchronization, which means synchronization can be achieved in a settling time, is desirable in some practical applications. However, most of the published results on finite time synchronization don’t include delays or only include discrete delays. In view of the fact that distributed delays inevitably exist in neural networks, this paper aims to investigate the finite time synchronization of memristor-based Cohen-Grossberg neural networks (MCGNNs) with both discrete delay and distributed delay (mixed delays). By means of a simple feedback controller and novel finite time synchronization analysis methods, several new criteria are derived to ensure the finite time synchronization of MCGNNs with mixed delays. The obtained criteria are very concise and easy to verify. Numerical simulations are presented to demonstrate the effectiveness of our theoretical results.


Introduction
Memristor, which was first proposed by Chua in 1971 [1], is deemed as the fourth fundamental circuit element besides inductor, capacitor and resistor. In 2008, the prototype of memristor was first realized by the scientists of Hewlett-Packard (HP) [2]. Memristor, the contraction of memory resistor, reflects the nonlinear relationship between charge and flux (see Fig 1). It has been proved that memristor has variable resistance and the function of memory. In the artificial neural network, the synapses are usually modeled by resistors [3]. Since memristors own memory and perform more like real biological synapses, now memristors have been utilized to replace the resistors in artificial neural network to build memristor-based neural network (MNN), which is the appropriate candidate for simulating the human brain [4].
On the other hand, synchronization of complex networks [5][6][7] has received much attention due to its great application prospect in many different fields such as image encryption [8], secure communications [9] and associative memory [10]. By utilizing a memristor to replace the diode in Chuas circuit, Chua obtained several oscillators in [11]. Since then, various memristive chaotic systems have been proposed by using the similar methods. As we know, chaos PLOS  controller used in [43] was very complicated. As far as we know, there has been no published result on finite time synchronization of MCGNNs with mixed delays until now. Inspired by the above analysis, this paper is devoted to studying the finite time synchronization problem of MCGNNs with mixed delays. The main contributions and originality of our paper are listed below: (i) This is the first attempt to investigate the finite time synchronization problem of MCGNNs with mixed delays, including time-varying discrete delays and distributed delays. Compared with the results in [43], the results in this paper are more general. (ii) The finite time synchronization analysis method used in this paper is a novel finite time synchronization analysis method, which has only been used in our another paper [44]. By adopting this novel analysis method, we derive some sufficient conditions that can ensure the finite time synchronization of the studied MCGNNs. Furthermore, the analysis method used in this paper can also be applied to analyze the finite time synchronization of other MNNs. (iii) In many literatures on the finite time synchronization of delayed systems, the controllers are very complicated. Although the neural network model considered in this paper is MCGNN with mixed delays, only simple feedback controllers are enough to derive the finite time synchronization of the studied MCGNNs. In some papers, the designed controllers were also similar to the controllers in this paper, however, only the asymptotical synchronization or the exponential synchronization of the studied systems can be obtained.
The rest of this paper is organized as follows. Some essential preliminaries are introduced in Section 2. In Section 3, our main results are derived. In Section 4, numerical simulations are presented to verify the theoretical results. Conclusions are drawn in Section 5.
Based on the relevant theories of differential inclusions and set-valued maps [47,48], we can derive that: where MCGNN (1) is referred to as the drive system, this is the corresponding response system: where R i (t) is the appropriate controller; the initial value of MCGNN (8) for i, j = 1, 2, . . ., n.

Main results
In this section, we will derive some sufficient conditions that can guarantee the finite time synchronization of MCGNNs (1) and (8). Theorem 1. Let assumptions A 1 -A 6 hold. If control gains p i and q i satisfy MCGNN (8) will be synchronized with MCGNN (1) in finite time under the controller (10). Proof. We design such a Lyapunov function: where je j ðzÞjdz; Finite time synchronization of MCGNNs By Lemma 1, the derivative of V 1 (t) can be calculated as: Based on assumptions A 2 and A 3 , it can be obtained that where θ 1 is between F À 1 i ðy i ðtÞÞ and F À 1 i ðx i ðtÞÞ, θ 2 is between y i (t) and x i (t). Based on assumptions A 2 and A 4 , it follows that signe i ðtÞ b ij ðtÞ½f j ðF À 1 j ðy j ðtÞÞÞ À f j ðF À 1 j ðx j ðtÞÞÞ j b ij ðtÞj Á l j jF À 1 j ðy j ðtÞÞ À F À 1 j ðx j ðtÞÞj b u ij l j w j je j ðtÞj: Similarly, we have signe i ðtÞ c ij ðtÞ½f j ðF À 1 j ðy j ðt À t ij ðtÞÞÞÞ À f j ðF À 1 j ðx j ðt À t ij ðtÞÞÞÞ c u ij l j w j je j ðt À t ij ðtÞÞj ð20Þ and signe i ðtÞ d ij ðtÞ Based on assumption A 5 , it follows that where γ i = 0 if e i (t) = 0, otherwise γ i = 1. Similarly, we get signe i ðtÞð c ij ðtÞ À c ij ðtÞÞf j ðF À 1 j ðx j ðt À t ij ðtÞÞÞ ðc ij À c ij ÞM j g i ð23Þ and On the other hand, Furthermore, since F À 1 i ðÁÞ is strictly monotone increasing, we know that signðF À 1 i ðy i ðtÞÞ À F À 1 i ðx i ðtÞÞÞ ¼ signe i ðtÞ. Then we have Calculating the derivatives of V 2 (t) and V 3 (t), we get that where assumptions A 1 and A 6 have been used.

Finite time synchronization of MCGNNs
Therefore, If the conditions in Theorem 1 are satisfied, we have where ε ¼ min By using the same analysis methods as those in [44], we can prove there exists a constant t Ã > 0 such that where e(t) = (e 1 (t), e 2 (t), . . ., e n (t)) T and k eðtÞ k 1 ¼ P n i¼1 je i ðtÞj: According to Definition 1, MCGNNs (1) and (8) achieve synchronization in finite time. The proof is completed. Remark 2. In Theorem 1, since the distributed delays in MCGNNs (1) and (8) are unbounded, it is difficult to estimate the settling time t Ã .
If the delay kernels satisfy where β ij > 0 are constants, i, j = 1, 2, . . ., n, MCGNN (1) can be written as In fact, MCGNNs (33) and (34) can also achieve finite time synchronization under the controller (10), what is more, the settling time t Ã can be estimated. Corollary 1. Let assumptions A 1 -A 5 hold. If control gains p i and q i satisfy MCGNN (34) will be synchronized with MCGNN (33) in finite time under the controller (10). Moreover, the settling time Proof. Consider such a Lyapunov function: where Referring to the proofs of Theorem 1 and Ref. [44], we can give the remaining proof of Corollary 1, which is omitted here.
Remark 3. In MCGNN (1), if R t À 1 K ij ðt À sÞf j ðx j ðsÞÞds is replaced by R t tÀ r ij ðtÞ f j ðx j ðsÞÞds, where 0 ρ ij (t) ρ ij , we can get a new MCGNN. Similarly to Corollary 1, we can prove that MCGNNs with this kind of distributed time-varying delays can achieve finite time synchronization under the controller (10), and the settling time t Ã can also be estimated.
Remark 4. It has been proved that controller (10) can synchronize MCGNNs effectively. Controller (10) consists of two parts: linear part −p i (η i (t) − ξ i (t)) and nonlinear part −q i sign Finite time synchronization of MCGNNs (η i (t) − ξ i (t)). In the proofs of Theorem 1 and Corollary 1, the nonlinear part of the controller is used to deal with the parameter mismatches of the drive-response MCGNNs, while the linear part of the controller plays a key role in driving the response MCGNN to synchronize with the drive MCGNN.
Remark 5. In [43], the authors also investigated the finite time synchronization of MCGNNs. However, the finite-time synchronization analysis methods they utilized were traditional ones [50], that is, they should prove _ V ðtÞ À aV Z ðtÞ, α > 0, 0 < η < 1, or _ V ðtÞ À aV Z ðtÞ þ yVðtÞ, α > 0, θ > 0, 0 < η < 1, where V(t) is the Lyapunov function. In this paper, we utilize some novel finite-time synchronization analysis methods [44]. First, we Then we use the strict mathematic analysis to derive the results. Moreover, though the delays considered in [43] were only discrete delays, the controller used in [43] was very complicated, i.e. R i ðtÞ ¼ À p i ðv i ðtÞ À u i ðtÞÞ À Z i signðv i ðtÞ À u i ðtÞÞ À P n j¼1 k ij signðv j ðtÞ À u j ðtÞÞ À P n j¼1 d ij signðv i ðtÞ À u i ðtÞÞjv j ðtÀ t j ðtÞÞ À u j ðt À t j ðtÞÞj. In this paper, we consider MCGNN model with mixed delays, however, the controller that we use is very simple, i.e.
). Remark 6. In MCGNN (1), if the memristive connection weights b ij (ξ i (t)) = b ij , c ij (ξ i (t)) = c ij and d ij (ξ i (t)) = 0, MCGNN (1) will reduce into the Cohen-Grossberg neural network model studied in [39,40]. Therefore, the theoretical results of this paper can be applicable to the Cohen-Grossberg neural networks in [39,40], while the opposite is probably not true. In this sense, the obtained results of this paper are less conservative.

Conclusion
This paper studies the finite time synchronization problem of MCGNNs with mixed delays. By utilizing some novel and effective analysis techniques, several sufficient conditions that can guarantee the finite time synchronization of MCGNNs with mixed delays are derived. The feedback controllers that we design are very simple, but they can solve the parameter mismatch problem of the drive-response MCGNNs perfectly. However, the conservativeness of the theoretical analysis probably makes the control gains of our feedback controllers much larger than those needed in the engineering applications. On the other hand, it is costly and impractical to control a network by applying controllers to all the nodes. Since adaptive pinning controller can avoid the high control gains effectively and reduce the number of the controlled nodes, our future work will focus on the synchronization control of MCGNNs via the adaptive pinning control. Numerical simulations are given to verify the obtained theoretical results.