Synchronization of reaction–diffusion Hopfield neural networks with s-delays through sliding mode control*

Xiao Liang mathlx@163.com

School of Mathematics and System Science, Shandong University of Science and Technology, China

Shuo Wang mathlx@163.com

School of Mathematics and System Science, Shandong University of Science and Technology, China

Ruili Wang

Institute of Applied Physics and Computational Mathematics, China

Xingzhi Hu

China Aerodynamics Research and Development Center, China

Zhen Wang mathlx@163.com

School of Mathematics and System Science, Shandong University of Science and Technology, China

Hopfield neural networks (HNNs) are intensively studied since it was first postulated in [10] due to their successful applications in numerous areas such as pattern recogni- tion,parallel computation, and associative memory [22]. The original version of the model is described by ODEs, which is just an approximation of real world. Several factors are neglected in this model.

To begin with, the delay is inevitably encountered in electronic implementation of neural networks (NNs) due to finite speed of switching and transmission of signals [8, 21] or deliberately introduced to deal with moving image processing [27]. It is necessary to incorporate distributed delays into the system when the NNs usually have a spatial extent due to presence of a multitude of parallel pathways with a variety of axon sizes and lengths [27]. After a scrutiny scan of published work on delayed HNNs, we find that most authors either concentrate on system with discrete delays or distributed delays independently. However, in the real signal propagation, we often encounter the case that NNs possess both discrete and distributed delays at the same time. From the viewpoint of mathematics, s-delays is an accurate and suitable tool to describe discrete and distributed delays at once since both of them can be included in the s-delays [9]. It has the form are activation functions, are Lebesgue–Stieljies measurable functions. So if we study the HNNs with s-delays, it means that our model is more general than the previous model.

On the other hand, diffusion phenomenon is also neglected in the original model of HNNs. Actually, diffusion effect cannot be ignored in NNs when electrons move in an in- homogeneous electromagnetic field [14,16]. Reaction–diffusion Hopfield neural networks (RDHNNs) not only have theoretical influence, but also have been used in numerous fron- tier regions such as image encription [32], pattern formation [40]. Compared with original HNNs, RDHNNs are described through partial differential equations with initial and boundary conditions. It not only involves the time variable, but also the space variables.

Based on above discussion, the model will be more exact if we study the diffusion phenomenon and delay effect simultaneously. However, it is worth noting that delay is a source of oscillation, bifurcation, and instability, which hinders the practical application of HNNs [8, 14, 21]. Meanwhile, diffusion can also harm the stability of the system [14, 16]. The dynamical behavior of HNNs will be even more complex when incorporating these two factors.

As an important collective behavior, synchronization of HNNs becomes a hot topic in recent decades due to their potential applications in secure communications, signal processing, distributed computation [7, 11, 17, 18, 28, 30, 31, 35, 36, 42]. It means that solution of drive system converges to the desired trajectory under appropriate control strategy [26]. However, synchronization of HNNs is still not fully conducted because it is hard to guide the solution so that it converges to ideal trajectory due to their complexity. This situation will be even worse if we take both reaction–diffusion phenomenon and s-delays into consideration.

Many effective strategies have been proposed for the synchronization of HNNs with either delay or reaction diffusion term. For example, the point-wise and optimal control by [31], pinning control by [17, 30], intermittent control by [7, 11, 18], impulse control by [28, 36, 42], nonlinear feedback control by [6]. It should be pointed out that these strategies heavily rely on Schur complement theorem, and in these papers, the criteria is free reaction–diffusion coefficients.

SMC is considered as a potential approach in synchronization of HNNs, which is a discontinuous control. Its main advantages are fast response, good transient perfor- mance, and robustness to external disturbance [1, 19, 23, 25, 29, 33, 34]. A great volume of the literature has been published on the theory and application of SMC for various systems [25, 33]. As to HNNs, [22] has pointed out the importance of sliding mode in recurrent neural networks, especially, how to prevent sliding. [29] investigates the synchronization of uncertain nonidentical chaotic neural networks with time delays via SMC. To the best of our knowledge, there is still no existing result on synchronization of delayed reaction– diffusion Hopfield neural networks via SMC until yet, let alone with s-delays.

Motivated by above discussion, synchronization of reaction–diffusion HNNs with s-delays is studied in this paper. Challenges and difficulties will be confronted since numerous factors are taken into account in this model. To begin with, both reaction–diffusion term and s-delay are considered in the model, and the delay is more general than previous ones. Moreover, reaction–diffusion term is an extension of Laplace operator . At last, we have checked the published work of SMC for distributed systems, there is still no simulation on the motion of equivalent control and SMC controller versus time and space.

Compared to some previously published results, our results are less conservative, and the model is more general. Main contributions are summarized as follows.

1. 1. Choosing appropriate Hilbert space for state variables is a critical step toward analyzing and approximating it. In the previous work of delayed reaction–diffusion HNNs, phase space is chosen to be the usual Euclidean space . The structure of is simple, concrete, and easy to grasp. However, the theoretical result is richer in abstract Hilbert space than that in , and the form of the system is much more concise in Hilbert space.
2. 2. SMC is successfully applied for the synchronization of this system. It is reliable and easy to be programmed and operated on the computer. Both sliding mode equation and control law are established by using the equivalent control. This is different from the previous work in SMC of other distribution systems, which use matrix splitting technique [19, 23, 34] or direct design of discontinuous control law [24,25].
3. 3. There is no existing specialized software suited for synchronization of reaction– diffusion Hopfield neural networks with s-delays. In order to accomplish these goals, we discrete this system and write the code by ourselves. Then simulation is obtained via Matlab to validate the efficiency of our result.
4. 4. The proofs of main theorems are based on constructing appropriate Lyapunov– Krasovskii functionals (LKFs). Meanwhile, some inequalities such as Young in- equality, Hanalay inequality are used in the process. These criteria are expressed in the matrix norm, which are also easy to check. These criteria are also explicitly expressed when compared with LMIs.

We list some notations, which will be used in the following sections.

• For , means is a positive definitive matrix;

• 

• is the Hadmard product betwen and [20];

• denotes space of square integrable fuctions on ;

• , it becomes a Hilbert space when eqquiped with usual inner product , and the corresponding norm is

• is the bananch space of continuos fructionals from , to with the supnorm

• 

• is the trace operator

• is called the Frobenius norm of

The drive system of reaction diffusion HNNs with s-delays is

where is the number of neurons, is the state vector. denotes the transpose of matrix is a connected bounded set with smooth boundary . Gradient operator of is a diagonal map with , represent activation function, is the rate matrix with is weights matrix. is bias vector. is the initial function, which is continuous in is time delay, is diffusion coefficient matrix, which is determined by Fick’s law [14]. Let is the Hadamard product of matrix and is the general divergence operator of matrix , which is defined asis the divergence operator of vector . Adiabatic boundary condition is used in this article.

S-delays are defined throught the Lesbesgue - Stieljies integral as are nondecreasing fuctions with bounded variation. In other words, there exist positive constants such that is the control strategy on drive system is the dimensionaless control matrix, wich is to be determinated

The response system of reaction diffusion HNNs with s-delays is

with is also continuous on . Other symbols have the same physical meaning as those in (1).

The tracking vector error is defined as the difference between the observed behavior of the drive system (1) and its desired behavior of response system (2), which means

From (1)–(3) we have

where

Remark 1. Let

where . Then, through calculating the Lebesgue–Stieljies integral, the governing equation of (1) is transformed to

This is the system with discrete delay.

If there exists the function such that , then calculating the Lebesgue–Stieljies integral, the governing equation of (1) is reduced to

This is the system with distributed delays.

2.1Tracking error in the Banach space

Let us define the diffusion operator as

and is the domain of , which is defined as [14]

Define the Nemytskii operator as follows [14]:

Then (4) is equivalent to the following functional differential equation in Hilbert space

where

In this paper, we assume

(H1)

(H2) There exist two positive constants such that .

Definition 1. The drive system (1) and response system (2) are said to be exponentially synchronized under appropriate controller . if there are constants and such that

where is defined in (3) and (6).

Let us construct a new matrix based on

Lemma 1. If (H2) holds and is a M-matrix, then .

Proof. We first prove the following equality by using the property of Hadmard product, and the basic relationship , denotes the standard inner product of Euclid space , then we have

where

Furthermore

where , which means

In other words, we have

Let in (9), and use the general Gauss formula for the matrix

where is the column of . By using the adiabatic boundary condition we have

Then using (H2), (9)–(10), Cauchy inequality , we have

where . Since is a M-matrix, then we have

The proof is complete

In this work, switching surface is defined as a linear combination of the current states

where satisfies , which will be determined later.

According to the SMC theory, when the system trajectories reach onto the switching surface, it follows that. In other words,

By substituting (6) into (14) we get

with the assumption that is invertible, we obtain the equivalent control

with

By substituting into (6) we get the sliding mode

where . We also assume that is invertible throughout this article.

For the convenience of study, we rewrite system (17) as follows:

We have the following main theorem of this article.

Theorem 1. If system (6)satisfies (H1).(H2) and

and is exchangeable, then the state vector of the drive system synchronizes to that of the response system on the sliding surface (13).

Proof. Let us define the Lyapunov–Krasovskii functional as follows:

The derivative of with respect to along any trajectory of system (18) is given by

By Lemma 1 we have

By the positiveness of diagonal entries of we have

were

By using in the sliding mode (18), the exchangeable assumption, we get

By Young inequality, [14], the definition of , where is a diagonal map, condition (H1), and total boundedness of Lebesgue–Stieljies integral we have

In this case we choose . By (19)(24) we have

with .

By the Hanalay inequality we have . By (H3) we have .

So the solution of (6) is exponentially stable on the sliding mode described by (18). By Definition 1 the drive system (1) and the response system (2) are synchronized in (13).

Remark 2. After a scrutiny scan of the latest works of synchronization for delayed or reaction–diffusion HNNs [2, 4, 7, 11, 12, 17, 18, 30, 37–40], we find that these criteria are expressed in the form of LMI toolbox, which heavily rely on the Schur complement theorem and optimization method. Compared with them, our criteria based on matrix norm are expressed explicitly. But their method is also an efficient tool, especially, when the uncertain disturbance is also taken into consideration [3, 4, 37, 40]. One of our object is to apply the LMIs techniques for this subject. Furthermore, our method is similar to the integral inequality method used in [41] and interval matrix method used in [36]. Although their criteria have wide range in application, the model in [41] and [36] is belong to an ordinary differential equation, only time variable is considered. We still hope their meth- ods can be extended to our model in the future work. By the way, the previous criteria for synchronization of reaction–diffusion HNNs are free reaction coefficients [7, 11, 17, 18,28,40]. But reaction coefficients is incorporated into our result through (H2) and (H3).

Theorem 2. Consider system (6)with assumptions (H1).(H3). Suppose that the switching surface is given by (13), the SMC law is designed to be

where is a positive scalar, which will be selected properly. Then switching surface s. is the sliding mode area under (25).

Proof. Consider the Lyapunov–Krasovskii functional as follows:

Using (13), the derivative of (26) with respect to time is given as follows:

By substituting SMC controller (25) into above equation we have

Since

we have

Then

We finally get

which means that any trajectory of (6) can be driven to remain on the sliding surface under SMC controller (25).

Remark 3. As we all know, chattering cannot be eliminated in the SMC. It is one of the shortages associated with this method. If we replace (25) with

the SMC controller becomes continuous, then chattering will disappear. However, the total robustness of SMC also will be lost correspondingly if continuation is utilized. So the reasonable strategy is to reduce rather than remove chattering in the future study.

Theorem 3. If the SMC law is chosen as (25), it can be shown that the reachability of the switching surfaceis guaranteed, and the reaching time satisfies

Proof. Let us construct the Lyapunov–Krasovskii functional as follows:

By using (25) derivate of with respect to is given as follows

By using

we have

We suppose that the trajectory of (6) will reach at time , and by definition of , which means

So

In other words, the SMC law (25) can drive error system (6) to the switching surface in finite time .

Remark 4. It can be seen from Theorems 1 and 3 that control law based on equivalent control is efficient for the target studied in this paper. The construction of control law is very tricky, and the method is direct. This is different from the previous work in SMC for other distribution systems, which use matrix splitting technique [19, 23, 34] or direct design of discontinuous control law [24, 25]. Their methods are suitable for those system, which is difficult to build equivalent control and can be utilized to deal with complex model. We will attempt to apply those methods to our system in the future study.

In this example,

Both of them are continuous in , and

and

Through calculating Leabesgue–Stieljies integral, we get

We also have by calculation are selected in this case.

Moreover, is a diagonal map. Since are global Lipschitz con- tinuous functions with. So (H1) is satisfied. We also have

such that , is a Laplacian operator.

The simulation is carried out through Matlab. The code is based on the finite difference method. Specifically, the second-order centered difference scheme is utilized to discrete the space. The Runge–Kutta–Chebyshev method is used to discrete the time. We use this scheme to simulate the dynamical behavior of uncontrolled part of drive system. For detailed information, please see Figs. 1–2. From Fig. 1 the surface of response system is very complicated, especially, it seems that there is no equilibrium for . It is unstable for them. To give a clear description of it, we also simulate the frequency of in Fig. 1, which coincides with Fig. 1.

We also design

Figure 1
Simulation of u1 (left) and u2 (right) in response system.rr

Figure 2
Frequency of u1 (left) and u2 (right) in response system

then

Since is chosen, then is the semipositive definite matrix,

This means . So (H3) is fulfilled. By Theorem 1 the behavior of (2) synchronizes to (1) in the switching surface .

According to (16), the equivalent control is

Figure 3
Simulation of e1 (left) and e2 (right) in error system.

Figure 4
Frequency of e1 (left) and e2 (right) in error system under SMC.

Furthermore, the SMC is defined as

when .

Figure 3 presents the asymptotic behavior for error system. The reason why we choose 3-D plot is that it is visual and intuitive. We can check the evolution of the system not only from the time span, but also from the space span. We can see that as time increases to the infinity, the error surfaces of and converge to the equilibrium 0 in the sliding manifold. It coincides with the result of Theorem 1.

To present a more clear information of , , we also give the trajectory of , for some chosen in Fig. 4, which coincides with Fig. 3.

We also give the simulation of equivalent control and SMC in Fig. 5. We can see that is slightly enhanced versus time to drive the error dynamics into the equilibrium 0.

Figure 5
Simulation of equivalent control veq (left) and SMC v (right).

This paper focuses on theoretical analysis of synchronization for reaction–diffusion Hopfield neural networks with s-delays, which is a prerequisite step for practical design of NNs. SMC is used in this process. Namely, linear switching surface is constructed and equivalent control involving delay and diffusion term is obtained in the Hilbert space . SMC is designed based on the equivalent control. We find that state vector of error system converges to the switching surface in finite time. Approximate time and sliding mode area is obtained by using the LKFs. Moreover, the exponentially stability of solution on the switching surface is also obtained by using Lyapunov–Krasovskii functional. These LKFs are different. Last, we provide an example to show the availability of our method, and the corresponding simulation is also given by using the Matlab. We can find that SMC is enhanced versus time to ensure the ideal property of controlled system.

SMC is still a promising field. Many different strategies have been proposed for SMC. For instance, integral sliding surface is a reasonable alternative for linear sliding surface. Unfortunately, after a scrutiny scan into published results on SMC for distributed systems, we find that the linear sliding surface is still the only option in selecting switching surface due to its distinct structure [1, 13, 19, 25]. So exploiting nonlinear sliding surface is one of our research direction in the future. The splitting technique will also be discussed for the design of SMC for distributed systems in the future work [19]. By the way, the noise is unavoidable in the real world [14, 15], so synchronization of stochastic reaction diffusion HNNs with s-delays will be considered in the next work. LMIs are also potential tools in the synchronization of this system, which are easy to be examined through Matlab toolbox by finding feasible solutions [5].

We would like to acknowledge our debt to the rigourous editors and preeminent reviewers. The paper is dramatically improved after following their constructive suggestions and detailed comments.