Exponential Antisynchronization Control of Stochastic Memristive Neural Networks with Mixed Time-Varying Delays Based on Novel Delay-Dependent or Delay-Independent Adaptive Controller

The global exponential antisynchronization in mean square of memristive neural networks with stochastic perturbation andmixed time-varying delays is studied in this paper. Then, two kinds of novel delay-dependent and delay-independent adaptive controllers are designed. With the ability of adapting to environment changes, the proposed controllers can modify their behaviors to achieve the best performance. In particular, on the basis of the differential inclusions theory, inequality theory, and stochastic analysis techniques, several sufficient conditions are obtained to guarantee the exponential antisynchronization between the drive system and response system. Furthermore, two numerical simulation examples are provided to the validity of the derived criteria.


Introduction
Memristor is a nonlinear resistor with memory function.
Because of the nonlinear nature, it has been found that memristor responds well to the variable strength of synapses in the human brain.Therefore, we use memristor instead of resistor to reform new neural networks model named memristive neural networks (MNNs).In recent decades, dynamic behaviors analysis of MNNs has been attracting increasing attentions [1][2][3].
As a typical dynamic behavior, the stability and synchronization problems of MNNs have been widely discussed, including exponential synchronization [4,5], lag synchronization [6], finite time synchronization [7,8], and antisynchronization [9,10].In communication system, the digital signal is transmitted by switching back and forth continuously between synchronization and antisynchronization, which strengthen security and secrecy.Besides that, the antisynchronization analysis of MNNs can provide the designers with some amazing properties, extensive flexibility, and opportunities.Thus, there are a few articles dealing with the antisynchronization issues of MNNs [11][12][13][14][15].In [11], they studied the exponential antisynchronization in mean square of MNNs with the uncertain terms which include nonmodeled dynamics with boundary and stochastic perturbations.By an intermittent sampled-data controller [12] and utilizing matrix measure approach and Halanay inequality [13], the antisynchronization control of MNNs with time-varying delays has been addressed.Wang et al. discussed the antisynchronization control of MNNs with multiple proportional delays [14] and mixed time-varying delays [15].Recently, the antisynchronization control has been widely employed in many fields, for example, secure communication, harmonic oscillation generation, and information science [16][17][18].
In practice, because of certain switching speeds of the amplifiers, time delays are often encountered in hardware implementation [19], which may affect the stability of the system, even result in oscillation, divergence, and instability phenomena.Consequently, a great deal of effort has been devoted to the stability analysis of MNNs with various types of time delays [20][21][22].Moreover, since an amount of parallel pathways of multiple axon sizes and lengths reside in the MNNs, such a unique nature can be appropriately modeled by distributed delay which means the signal transmission is distributed over a specific period of time.Hence, the authors in [23][24][25] have concentrated on the mixed delays.Meng and Xiang [24] considered a class of recurrent MNNs with mixed time-varying delays, in which the passivity and exponential passivity were investigated.
Furthermore, in [25], Yang et al. studied the th moment exponential stochastic synchronization for MNNs with discrete and distributed time-varying delays.However, the discrete delay () in [25] must be differentiable.Clearly, such a constraint on the delay term () was relatively strong.So, we discuss two kinds of discrete time-varying delay in this paper.One is that its derivative is bounded; the other is that it is not differentiable, or the derivative is unknown or unrestricted.
On the other hand, as a result of random fluctuations from the release of neurotransmitters or other probabilistic causes in the nervous system, synaptic transmission is indeed a process accompanied by noise.The stability analysis of MNNs with stochastic perturbation has aroused great interests of many researchers [26][27][28].For example, Song and Wen [27] investigated the stochastic MNNs with mixed delays and proposed the exponential synchronization criteria in the th moment.Subsequently, Bao et al. [28] discussed a class of coupled stochastic MNNs with probabilistic timevarying delay in order to achieve exponential synchronization.However, to the best of our knowledge, the research on the exponential antisynchronization analysis of stochastic MNNs with mixed delays is still an open problem.
Motivated by the above discussion, we focus our minds on the exponential antisynchronization problem of the MNNs with stochastic perturbation and mixed time-varying delays.Compared with other existing articles [11,28,29], our model is more complex and closer to the actual system; the obtained results are less conservative.The main contributions of this paper can be summarized in the following: (1) The presented MNNs model contains not only stochastic perturbation but also two types of timevarying delays, namely, discrete and distributed timevarying delays.
(2) It is known that the time delays have a great influence on the stability of the MNNs, so the time delays cannot be neglected.As far as we know, many articles are based on the assumption that the time delay is bounded and derivable.Actually, it may happen that the time delay () is not differentiable or its derivative is unknown or unrestricted.Therefore, we study two kinds of discrete time-varying delay.
(3) Under two new types of controllers with delaydependent and delay-independent, the obtained criteria which need no excessive numerical calculation can be extended to other general MNNs models.
The rest structure of this paper is outlined as follows.In Section 2, the models of the stochastic MNNs with mixed time-varying delays and some preliminaries are introduced.In Section 3, the main results on exponential antisynchronization of the stochastic MNNs are derived.In Section 4, some numerical simulations are presented to demonstrate the efficiency of the theoretical results.In Section 5, the conclusion is showed.

Model Description and Preliminaries
On the basis of the above discussion, we propose a class of MNNs with discrete and distributed time-varying delays described by the following differential equations: where  corresponds to the number of neurons in system (1).
Remark 2. Compared with the articles already published [28,29], the proposed system contains not only discrete timevarying delay () but also distributed time-varying delay (), while the stochastic perturbation is also taken into account.Therefore, the obtained results have a high value of practical application in this paper.
As a matter of convenience, we will take advantage of the following assumptions.
Assumption 4.   š  + ×  ×  →  is locally Lipschitz continuous and meets the linear growth condition with   (, 0, 0) = 0.There exist nonnegative constants   ,   , such that, for ,  = 1, 2, . . ., , Assumption 5.For ,  = 1, . . ., , Remark 6.In [4], however, the error system was defined based on the following assumption: In fact, this assumption has been proved not always to be correct in [5].Recently, many researchers have tried to explore a novel and appropriate way to solve this problem.So far, there are two kinds of convincing method.One is, in [31], according to the switching parameter Φ  ; the authors discussed the parameter in four cases and obtained several synchronization conditions of the chaotic MNNs with time delays.The other is, in [11,[15][16][17], by Assumption 5; the authors turned to study the antisynchronization issues of MNNs.
Lemma 7.For the stochastic system, where () is the Wiener process and it is obvious that () = 0. L(, ()) is the operator defined as follows: where Lemma 8 (see [32]).If  and  are real matrices which have appropriate dimensions, then there exists a constant  > 0, such that Lemma 9 (see [33], Jensen's inequality).For any constant matrix  > 0, a scalar  ≤ , and a vector function (): [, ] →   , then the following inequality holds: Definition 10.The error system () with mixed delays and stochastic perturbations can exponentially converge to zero in mean square if there exist positive constants  > 0 and ] > 0, such that {‖()‖ 2 } ≤  (−]) , for  ≥ 0, where ] is called the exponential convergence rate.Systems ( 6) and ( 8) are said to be exponential antisynchronization in mean square sense.

Main Results
In this section, we get some sufficient criteria to achieve the exponential antisynchronization of the stochastic MNNs with mixed time-varying delays.Then two corollaries are also derived for the stochastic MNNs.The novel delay-dependent adaptive controller   () is designed as follows: where the constants   > 0,   > 0, for  = 1, 2, . . ., .
For convenience, some notations are given.Let Theorem 11.Under Assumptions 3, 4, 5, and 13, systems (6) and ( 8) achieve exponential antisynchronization in mean square sense under the delay-dependent adaptive controller (23).For a given constant ] > 0, if there exist constants   > 0,   > 0 such that the conditions hold then error system (15) can be exponentially converged to zero.
Proof.We choose the following Lyapunov functional: where Then we have By Itô differential formula, we obtain the stochastic derivative of () in the form of From Lemma 7, it is clear that Moreover, from mean value inequality and Assumption 3, we have Mathematical Problems in Engineering Under Assumption 4, one has then we obtain the following estimation: Let  Assumptions 3,4,and 13,system (15) will converge to zero.For a given constant 0 < ] < 2  −  , if there exist constants   > 0,   > 0 and the conditions hold ) systems ( 6) and ( 8) are exponentially antisynchronized in the mean square sense under the following controller: Proof.We consider the following Lyapunov function: Combining with Assumption 13, the proof of Corollary 14 is analogous to Theorem 11, so we omit it here.
Remark 15.Based on the assumption that the delay () is bounded and differential, Theorem 11 provides a delaydependent adaptive controller.It is worth mentioning that the criteria need no excessive numerical calculation such as solving LMIs [11] or computing complex algebraic conditions [28].In Corollary 14, the delay () is not required to be differential but only bounded which makes our result more general.Further, there is an upper limit (2  −   ) of the exponential convergence rate ].When the exponent rate ] exceeds the upper limit, system (15) may not be able to converge or even cause unpredictable result.Let The vector form of system ( 15) is given by Moreover, we impose the following assumption and some lemmas.
If there is no time delay item, the delay-independent adaptive controller can be described as follows: Next, we will discuss the antisynchronization of systems ( 6) and ( 8) under the above delay-independent adaptive controller ().
Proof.We construct the following Lyapunov function: where Then we have According to Lemma 7, we get that ×  (,  () ,  ( −  ()))] , It follows from Lemma 8 that Under Lemma 17, we obtain that From Assumption 16, it is clear that Mathematical Problems in Engineering Then we have ℎ ( ()) ) . (57) Let Considering the condition of Corollary 14, we get According to Definition 10, this implies that the exponential antisynchronization of systems ( 6) and ( 8) is achieved.Thus, the proof is completed.
When there is no distributed time-varying delay in system (45), the error system is introduced as follows: (62) Corollary 19.Assumptions 3,4,and 16 hold, for a given constant ] > 0; if there exist matrix Π > 0 and constant  > 0 such that the conditions hold system (62) will exponentially converge to zero with the action of the delay-independent controller (48).
Proof.We construct the following Lyapunov function: ]   ()  () . (64) According to the proof of Theorem 18, Corollary 19 is not difficult to obtain.Hence, it is omitted here.
Remark 20.Theorem 18 holds only when some special conditions are fulfilled: (1) all parameters meet some certain conditions; (2) the novel delay-independent adaptive controller which is much easier to implement in practice is provided.
What makes that all the more remarkable is that there is no upper bound constraint on the exponent convergence speed ] in Theorem 18.Then we can improve the exponent convergence speed ] appropriately within the permissible range to achieve synchronization as soon as possible.So Theorem 18 is less conservative from a certain perspective.

Numerical Simulation
In this section, several numerical examples are offered to illustrate the effectiveness of the results obtained in the above section.
When we ignore the impact of the distributed delay, namely, () ≡ 0, system (72) will be rewritten as  Similarly, all the parameters are the same as those from above.According to Corollary 19, systems (77) and (78) under controller (76) achieve the exponential antisynchronization in the mean square sense.Define error system () = () + (); Figure 4 depicts the curves of error systems without or with controller (76).It is obvious that Figure 4 proves the error system converges quickly to zero.

Conclusion
In this paper, we addressed the antisynchronization issues for a class of MNNs with stochastic perturbation and mixed delays, including discrete and distributed time-varying delays.Two novel adaptive controllers were designed on the basis of delay-dependent and delay-independent to ensure that the drive system can achieve the exponential antisynchronization with the response system in spite of their initial conditions.Through the use of the Lyapunov stability method, inequality analysis technique, and the differential inclusion theory, we proposed two types of exponential antisynchronization criteria for these systems.Moreover, the obtained criteria need no excessive numerical calculation.Some numerical examples have been shown to validate the effectiveness of our theoretical results.

Figure 4 :
Figure 4: (a) Curves of error systems without controller; (b) curves of error systems with delay-independent controller (76).
Curves of error systems without controller; (b) curves of error systems with delay-dependent controller (71).