Robust Stability of Nonlinear Diffusion Fuzzy Neural Networks with Parameter Uncertainties and Time Delays

Rao, Ruofeng; Tang, Gaozhi; Gong, Jiuqi; Wan, Xiaoyan; Wu, Guanghong; Zhang, Qiao; Zhong, Shouming

doi:https://doi.org/10.1155/2018/6263931

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Abstract Introduction Preliminaries Conclusions Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2018 | Article ID 6263931 | https://doi.org/10.1155/2018/6263931

Robust Stability of Nonlinear Diffusion Fuzzy Neural Networks with Parameter Uncertainties and Time Delays

Ruofeng Rao,¹Gaozhi Tang,¹Jiuqi Gong,¹Xiaoyan Wan,¹Guanghong Wu,¹Qiao Zhang,¹and Shouming Zhong²

Academic Editor: Michele Scarpiniti

Received23 Oct 2017

Revised11 May 2018

Accepted24 May 2018

Published16 Jul 2018

Abstract

In this paper, a class of nonlinear p-Laplace diffusion BAM Cohen-Grossberg neural networks (BAM CGNNs) with time delays is investigated. In the case of with , the authors construct novel Lyapunov functional to overcome the mathematical difficulties of nonlinear p-Laplace diffusion time-delay model with parameter uncertainties, deriving the LMI-based robust stability criterion applicable to computer MATLAB LMI toolbox and deleting the boundedness of the amplification functions. And in the case of , LMI-based sufficient conditions are also inferred for robust input-to-state stability of reaction-diffusion Markovian jumping BAM CGNNs with the event-triggered control, which is different from those of many previous related literature. In particular, the role of diffusion can be reflected in newly acquired criteria. Finally, numerical examples verify the effectiveness of the proposed methods.

1. Introduction

In recent decades, reaction-diffusion neural networks have been the subject of research due to the fact that electrons have diffusion behaviors in an inhomogeneous magnetic field, and the role of diffusion items have always been investigated and discussed in many existing results ([1–4]). Since the conduction velocity of electrons and components is limited, the phenomenon of time delays inevitably appears in various practical projects. Thereby, time-delay reaction-diffusion systems are relatively common objects of study. For example, in [5], the following time-delay reaction-diffusion Cohen-Grossberg neural networks (CGNNs) with impulse was studied (see [7, (7)]), where is Hadamard product of matrix and vector gradient (see [6] for details).

In [7], the stability of the following BAM Cohen-Grossberg neural networks (BAM CGNNs) with distributed delays was discussed.

The Cohen-Grossberg-type BAM neural network model was initially proposed by Cohen and Grossberg [8] in 1983. The model not only generalizes the single-layer autoassociative Hebbian correlator to a two-layer pattern matched heteroassociative circuit but also possesses Cohen-Grossberg dynamics, and it has promising application potentials for tasks of classification, parallel computation, associative memory, and nonlinear optimization problems. Since then, a lot of research has been done on BAM CGNNs models ([7, 9–11]). Besides, owing to biological engineering backgrounds and population dynamics, economics, physical engineering, and other reasons, the stability of nonlinear diffusion systems have received widespread attention [11–17]. For example, in [11], the author studied the following nonlinear diffusion fuzzy system, involved to time-delay BAM Cohen-Grossberg neural networks.

Under the complex conditions

and other conditions, a stability result ([11, Theorem 3.2]) was given, where

In recent years, some methods and ideas of related literature ([5–45]) inspire our current work. In this paper, we shall discuss the robust stability of nonlinear p-Laplacian diffusion Takagi-Sugeno (T-S) fuzzy system with discrete delays and distributed delays. Actually, T-S fuzzy models provide a successful method to describe certain complex nonlinear system using some local linear subsystems ([31, 32, 46]). Besides, there exist parameter errors unavoidable in factual systems due to aging of electronic components, external disturbance, and parameter perturbations. Therefore, the robustness of the system stability should be investigated, too. Our main objectives are as follows: (1)Changing (4) into linear matrix inequalities (LMIs) applicable to computer MATLAB LMI toolbox, which can be adapted to large-scale calculation in practical engineering.(2)Ensure that the nonlinear diffusion term plays a role in the LMI-based stability criterion while in some existing results ([6, Theorem 3.1], [18, Theorem 3.1], [19, Theorem 3.1],), the role of the nonlinear diffusion term was neglected in their LMI-based criteria.(3)Deleting the boundedness of amplification function in some existing results (see, e.g., [7, 9, 19, 21]).

For these purposes, we need to achieve the following works: (i)Improve [11, Lemma 3.1] and make it adopted to LMI-based criterion, in which the nonlinear diffusion can play roles.(ii)Construct a novel Lyapunov functional and design comprehensive applications of variational method, Young inequality, and LMI technique so that LMI-based criterion can be derived for the nonlinear diffusion fuzzy system with parameter uncertainties, discrete delays, and distributed delays.(iii)Relax the restrictions of amplification function so that the boundedness of is not necessary. At the same time, employing LMI technique guarantees structuring LMI-based criterion.(iv)Explore the input-to-state stability of reaction-diffusion Markovian jumping BAM CGNNs with time delays and the event-triggered control

For convenience’s sake, we still need to introduce some standard notations: (i): a nonnegative (nonpositive) matrix, that is, for all (ii): represents the matrix C = (A − B) satisfying C ≥ 0 (≤4 0).(iii): a positive (negative) definite matrix.(iv): a nonnegative (nonpositive) definite matrix.(v): this means is a nonnegative (nonpositive) definite matrix.(vi): this means is a positive (negative) definite matrix.(vii) and denote the largest and smallest eigenvalue of matrix , respectively.(viii)Denote for any matrix ;(ix) for any vector .(x) implies for all (xi) implies for all where and (xii)I: identity matrix with compatible dimension.(xiii)The Sobolev space (see [28] for details).(xiv)Denote by the lowest positive eigenvalue of the boundary value problem (see [28] for details)

2. Preliminaries

Consider the following Takagi-Sugeno fuzzy p-Laplace partial differential equations with distributed delay.

Fuzzy rule :

If is and is then where is the premise variable and is the fuzzy set that is characterized by membership function. is the number of the if-then rules, and is the number of the premise variables. and are diffusion coefficients matrices. is Hadamard product of matrix and (see e.g., [13] for details) and so is . Let be a given scalar, and be a bounded domain with a smooth boundary of class by . Denote and . For any given is the state variable of the th neuron at time in space variable and so is . , and , in which is the neuron activation function of the th unit of time in space variable and so is . Both and are discrete time delays with and . And distributed delays and with and . In addition, the positive scalar . Here, , and all may be . Besides, there is a positive scalar l₀< 1 such that and for all . , and , in which represents an amplification function and so does . , and , in which both and are appropriately behavior functions. and are connection weight strength coefficient matrices, and and are real-valued matrix functions which represent time-varying parameter uncertainties.

By means of a standard fuzzy inference method, (7) can be inferred as follows, where , , and is the membership function of the system with respect to the fuzzy rule . can be regarded as the normalized weight of each if-then rule, satisfying and .

Particularly in the case of , the system (8) is the so-called reaction-diffusion impulsive Markovian jumping BAM Cohen-Grossberg neural networks (BAM CGNNs). Inspired by some methods and conclusions of some related literature ([47–51]), we shall discuss the input-to-state stability reaction-diffusion BAM CGNNs with the event-triggered control in Section 4, for seldom existing literature involved to such complex model with feedback control.

Lemma 2.1. , , and .
Note that Lemma 2.1 is the particular case of the famous Young inequality.

Lemma 2.2 (Schur complement [52]) Given matrices , , and with appropriate dimensions, where and , then if and only if or where , , and are dependent on .

3. Robust Stability on Nonlinear p-Laplacian Diffusion System in the Case of

Throughout this paper, we assume that and , where we denote and for short. In addition, we always denote and . Denote by and by and so do and .

Lemma 3.1. Let be a positive real number, and a positive definite matrix. Let and be a solution of (8). Then we have

Proof. Since u is a solution of (8), it follows by Gauss formula and the Dirichlet zero boundary condition that

Another inequality can be similarly proved. And so the proof is completed.

Remark 1. Lemma 3.1 improves [11, Lemma 3.1] and [18, Lemma 2.3] for the first time, which makes a contribution to the final LMI criterion.

Remark 2. In the case of or , the first eigenvalue (see, e.g., [28]).

Remark 3. If and , the first eigenvalue (see, e.g., [26]).

In this section, we suppose (H1)There exist positive definite matrices , , , and such that where and .(H2)There exists positive definite matrices and such that and where and .(H3)There are positive definite matrices and such that where and .

Remark 4. The condition (H1) implies that the boundedness of amplification functions and are unnecessary in the case of with , for we may take , which is actually unbounded for . Below, we denote for convenience where , , , and are diagonal matrices.

Theorem 3.2. Suppose that the conditions (H1)–(H3) hold and with being an even number and being an odd number. Besides, there are four nonnegative matrices , , , and such that Assume, in addition, and there is a positive definite matrix such that and then there exists the globally asymptotically robust stable unique equilibrium point for (8).

Remark 5. Condition (4) does not complete the matrix form. However, (19)–(20) are complete linear matrices inequalities, which have even gotten better at dealing with the calculation of the large operations involved in the practical engineering by way of computer MATLAB programming.

Proof. There are three steps to the proof.

Step 1. We claim that the null solution is the unique equilibrium point for (8).

In fact, we know from (H2)–(H3) that , and hence and are the equilibrium solution of (8).

Moreover, we prove that the equilibrium point is unique. Indeed, it follows from (H1) that . Let (21) be an equilibrium point for (8) then we get

If (23) is another equilibrium point of (8) we can actually deduce from (22) that and then

Similarly,

Combining (25) and (26) implies and (18) yields or

Thereby, the null solution is the unique equilibrium point for (8).

Remark 6. In ordinary differential systems, the uniqueness of the equilibrium solution can be determined by the existence of the equilibrium solution and its global asymptotic stability. However, (8) is a partial differential system, including two different variables: and . Since the existence of the equilibrium solution and its global asymptotic stability only determines the equilibrium solution which is unique about variable , but it may be not unique on variable . Hence, it is necessary to verify the uniqueness of the equilibrium solution.

Step 2. To derive LMI-based criterion in which the nonlinear diffusion terms can play roles, we need to construct new Lyapunov-Krasovskii functionals as follows: where

Remark 7. The uncertainty of parameters brings a difficulty to design the Lyapunov functions. If imitating the previous Lyapunov functions in existing literature, for example, let one can find it impossible that the sufficient conditions of stability criterion can be derived. In addition, Lyapunov functions (33) and (34) help us to derive the complete linear matrix inequality condition for the stability criterion of nonlinear diffusion system (8).

Step 3. We claim that the null solution is globally asymptotically robust stable.

Evaluating the time derivation of along the trajectory of the (8), we can derive from Lemma 3.1where we simply denote and

It follows by (H1), (H2), and the conditions on the parameter that

So combining (H1), (H2), and (39) results in

From Lemma 2.1, (H1), and (H3), we get where .

Besides, we can conclude from (H2), (H3), and Lemma 2.1 that where .

So we have

Besides, we get by (32)

One can deduce from (33) that

Hence,

Similarly, we can deduce from , , and that

Therefore, (17), (19), and (20) yield

It follows by the standard Lyapunov functional theory that the null solution of (8) is globally asymptotically robust stable. And the proof is completed.

Remark 8. There have been some other approaches removing boundedness of amplification functions. For example, in [53], an appropriate Lyapunov-Krasovskii functional is set up to derive the LMI-based -stability for discrete time-delay system. This is really a good result. However, in this paper, our system model (8) is the continuous system different from the discrete system ([54, (1)]). Of course, the main results of this paper are inspired by some methods and ideas of these documents.

Remark 9. The Lyapunov functionals (33) and (34) are similar to the quadric form different from those of [11, 13, 14, 20]. Actually, the quadric form and matrix form help us to derive the LMI-based criterion.

Remark 10. The boundedness of amplification functions and may be unbounded while amplification functions are always proposed to be bounded in many existing results (see, e.g., [7, 9, 18–21]).

If the diffusion phenomena are ignored, (8) degenerates into the following BAM CGNNs with discrete and distributed time-varying delays:

Since in ordinary differential systems, the uniqueness of the equilibrium solution can be determined by the existence of the equilibrium solution and its global asymptotic stability, and the diffusion items disappear, we can directly deduce the following corollary from Theorem 3.2:

Corollary 3.3. Suppose that the conditions (H1)–(H3) hold. Besides, there are four nonnegative matrices , and such that and there is a positive definite matrix such that and then there exists the unique globally asymptotically robust stable equilibrium point for (49).

Remark 11. For the BAM CGNNs (53), Corollary 3.3 deletes the boundedness of amplification functions and , improving related results (see, e.g., [7, 9, 18, 19, 21]). This is also shown below (Table 1).

4. Input-to-State Stability of Markovian Jumping Reaction-Diffusion BAM CGNNs with Event-Triggered Control in the Case of

In this section, we consider the following Markovian jumping reaction-diffusion BAM CGNNs with event-triggered control under Dirichlet zero-boundary value. for all , the initial value is , , and , where represent feedback, and represent the unknown exogenous disturbance of the neuron. For any the time is the triggering time or update time. Between the triggering times and , the feedback control is designed as where , , , and . Here and are constants for all .

Let be the given probability space where is the sample space, is , the algebra of subset of the sample space, and is the probability measure defined on . Let and the random form process be a homogeneous, finite-state Markovian process with right continuous trajectories with generator and transition probability from mode at time to mode at time , , and . where is transition probability rate from to and and .

Let and be the error signal defined by then we actually get

Define the event-triggering mechanism by where , , and .

Remark 12. Such that is always defined well on many occasions. For example, let initial value , , and , then we must get .

In this section, we assume that the conditions (H1)–(H3) hold still in the case of .

Besides, suppose that (H4) with a positive definite matrix such that(H5) with a positive definite matrix such that

For any mode , which do not have to be diagonal matrices or other special matrices.

In addition, we assume that which can guarantee that , and is a trivial solution of (53).

Besides, there are nonnegative matrices and such that

Before giving the man result of this section, we need the following lemma:

Lemma 4.1 ([54]) Let , , and . Then we have

Definition 4.2. System (53) is called robust stochastic input-to-state in mean square stable for all admissible uncertainties satisfying (63), if for , there exist function and such that where is continuous strictly increasing with with , for each fixed is decreasing for fixed and .

Theorem 4.3. Assume the conditions (H1)–(H5) hold. Suppose that there is a sequence of positive definition matrices and positive scalars , , , and such that the following LMI conditions hold for any mode : where represents the identity matrix with suitable dimension under different cases for convenience. If, in addition, then (53) is a robust stochastic input-to-state stable in mean square.

Proof. Construct the Lyapunov-Krasovskii functionals as follows: where each is positive definition diagonal matrix.Due to we get and where and .
Similarly, we get Next, Let be the weak infinitesimal operator, then we get Similarly, Hence, we get That means where and In addition, for any , the definition of derives So we get or where (66) and Schur complement lemma yield that and hence with and .
Furthermore, Dynkin’s formula yields where Applications of the Comparison principle to (84) reaches which derives which together with Definition 4.2 implies that (53) is robust stochastic input-to-state stable in mean square.

Remark 13. Theorem 4.3 provides a new stability criterion which is different from the existing criteria of [55–59]. In addition, to the best of our knowledge, it is the first time to investigate input-to-state stability of reaction-diffusion time-delay system with event-triggered control. Especially, the diffusion items play roles in the criterion.

5. Numerical Examples

Example 5.1. Consider (8) with the following parameters: , , , and then the first eigenvalue (see Remark 2).

Let , , , , , , , , , , , , , , and , ,

Let , and

So we can use MATLAB software to compute (18), obtaining

Moreover, utilizing MATLAB LMI toolbox to solve LMIs (19)–(20) reaches the feasibility data as follows:

Now, one can conclude from Theorem 3.2 that there exists the globally asymptotically robust stable unique equilibrium point for (8).

Remark 14. From Table 1, we know, our new results (Theorem 3.2 and Corollary 3.3) is novel because the boundedness of amplification functions becomes unnecessary.

Remark 15. From Table 2, we know, our Theorem 3.2 is novel, different from those of existing results.

Example 5.2. Consider (63) with the following parameters: and , , and . And so (see Remark 3).

Let , , , , , , , , , , , , ,

Let , and

Let , , and and then we can compute and verify that (68) is satisfied. Now using computer MATLAB LMI-toolbox to solve LMI (66) gives the feasibility data as follows:

According to Theorem 4.3, (53) is robust stochastic input-to-state stable in mean square.

Remark 16. This paper is inspired by the methods and conclusions of the previous literature [55–59]. But the sufficient conditions of Theorem 4.3 is easier to be verified than those of existing results.

6. Conclusions

In this paper, we mainly provided two novel conclusions for p-Laplace diffusion BAM CGNNs. In the case of with , the authors construct novel Lyapunov functional to overcome the mathematical difficulties of nonlinear p-Laplace diffusion time-delay model with parameter uncertainties, deriving the LMI-based robust stability criterion applicable to computer MATLAB LMI toolbox, deleting the boundedness of the amplification functions. On the other hand, when , LMI-based sufficient conditions are also inferred for robust input-to-state stability of reaction-diffusion Markovian jumping BAM CGNNs with the event-triggered control, which is different from those of many previous related literature. As far as we are concerned, seldom literature involved a reaction-diffusion stochastic system with time delays and the event-triggered control. It is the first time to explore the method for the stability analysis of this system. Finally, numerical examples illustrate the effectiveness and feasibility via computer MATLAB LMI toolbox.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

All authors typed, read, and approved the final manuscript.

Acknowledgments

This research was supported by the 2018 teaching reform project (2018JG38) of Chengdu Normal University titled “Financial mathematics course – Extended applications of stochastic process in dynamical system”, the National Natural Science Foundation of China (Grant nos. 61771004 and 61533006), Scientific Research Fund of Science Technology Department of Sichuan Province (2012JY010), and Scientific Research Fund of Sichuan Provincial Education Department (18ZA0082, 14ZA0274,12ZB349, and 08ZB002).

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Copyright © 2018 Ruofeng Rao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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