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Xiongrui Wang, Ruofeng Rao, Shouming Zhong, "New Stability Criterion for Takagi-Sugeno Fuzzy Cohen-Grossberg Neural Networks with Probabilistic Time-Varying Delays", Mathematical Problems in Engineering, vol. 2017, Article ID 3793157, 11 pages, 2017. https://doi.org/10.1155/2017/3793157
New Stability Criterion for Takagi-Sugeno Fuzzy Cohen-Grossberg Neural Networks with Probabilistic Time-Varying Delays
A new global asymptotic stability criterion of Takagi-Sugeno fuzzy Cohen-Grossberg neural networks with probabilistic time-varying delays was derived, in which the diffusion item can play its role. Owing to deleting the boundedness conditions on amplification functions, the main result is a novelty to some extent. Besides, there is another novelty in methods, for Lyapunov-Krasovskii functional is the positive definite form of powers, which is different from those of existing literature. Moreover, a numerical example illustrates the effectiveness of the proposed methods.
Cohen-Grossberg neural networks (CGNNs) have many practical applications, like artificial intelligence, parallel computing, image processing and recovery, and so on ([1–6]). But the success of these applications largely depends on whether the system has some stability, and so people began to be interested in the stability analysis of the system. In recent decades, reaction-diffusion neural networks have received much attention ([7–13]), including various Laplacian diffusion ([6, 14–20]). Besides, people are paying more and more attention to fuzzy neural network system ([21–34]), due to encountering always some inconveniences such as the complicity, the uncertainty, and vagueness ([27, 35–37]). For example, in , Zhu and Li investigated the following fuzzy CGNNs model:
In , Muralisankar and Gopalakrishnan studied the following T-S fuzzy neutral type CGNNs with distributed delays: Besides, Balasubramaniam and Syed Ali discussed Takagi-Sugeno fuzzy Cohen-Grossberg BAM neural networks with discrete and distributed time-varying delays in .
Note that there is the following bounded condition on amplification functions in many literatures (see, e.g., [38, Theorem ]) related to CGNNs:
So, in this paper, we try to delete this bounded condition on amplification functions. This is the main purpose of this paper.
Consider the following fuzzy Takagi-Sugeno -Laplace partial differential equations with distributed delay.
Fuzzy Rule . IF is and is THENwhere is an arbitrary open bounded subset in . is the premise variable and is the fuzzy set that is characterized by membership function. And is the number of the IF-THEN rules; is the number of the premise variables. , where is the state variable of the th neuron and the th neuron at time and in space variable . Matrix with each , and is diffusion operator. denotes the Hadamard product of matrix and (see  for details). Matrices and , where and represent an amplification function at time and an appropriate behavior function at time . is the connection matrix. Time delays . is the activation function of the neurons. And the second and third equations of (4) imply the initial condition and the Dirichlet boundary condition, respectively.
By way of a standard fuzzy inference method, (4) 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 .
Next, we consider the following information for probability distribution of time delays : Here the nonnegative scalar . Define a random variable as follows: So, in this paper, we consider the following Takagi-Sugeno (T-S) fuzzy system with probabilistic time-varying delays:
System (8) includes the following integrodifferential equations:
Particularly when , system (8) degenerates into the so-called reaction-diffusion CGNNs:
Throughout this paper, we assume with being even number and being odd number. Besides, suppose that the following conditions hold:(H1)There exist positive definite matrices and such that where and .(H2)There exists a positive definite matrix such that and (H3)There is a positive definite matrix such that
From (H1)–(H3), we know that and is an equilibrium of fuzzy system (8).
Remark 1. There are numerous functions satisfying (H1). For example, if , we may set It is obvious that So the function is unbounded for Moreover, One can know from (16) that with and .
Remark 2. The amplification function defined as (7) is actually unbounded for . However, various bounded conditions always imposed restrictions on the amplification functions of existing literature ([3–6, 9, 10, 24, 27, 28]). Hence, our condition (H1) is weaker, which will make a corollary with regard to ordinary integrodifferential equations (9) become novel.
For convenience’s sake, we need to introduce the following standard notations similarly as :(i)The Sobolev space (see  for details).(ii)Denote by the lowest positive eigenvalue of the boundary value problem (see  for details).
Lemma 3. One has
Note that Lemma 3 is the particular case of the famous Young inequality.
3. Results and Discussion
Lemma 4. Let be a positive definite matrix and be a solution of the fuzzy system (8). Then one has where , , and is a positive scalar, satisfying .
Proof. Since is a solution of system (8), it follows by Gauss formula and the Dirichlet zero-boundary condition that
Theorem 6. If there exists a positive definite matrix and two positive scalars , such that the following inequalities hold: then the null solution of fuzzy system (8) is globally asymptotically stable, where matrices , , , and .
Proof. Firstly, we can conclude from (H1)–(H3) that is an equilibrium point for system (8).
Next, consider the Lyapunov-Krasovskii functional: where Here, is a solution for stochastic fuzzy system (8). Below, we may denote by and by for simplicity.
Remark 7. It is obvious that our Lyapunov-Krasovskii functional is the positive definite form of powers, which is different from those of existing literature ([41–43]). For example, in , the model is also neural networks with discrete time delay and distributed delays: In [42, Theorem ], the corresponding Lyapunov-Krasovskii functional is as follows: which is the positive definite form of powers. And the conclusion of [42, Theorem ] is the asymptotical stability in the mean square, which is also similar to that of our Theorem 6. However, by means of our Lyapunov-Krasovskii functional with the positive definite form of powers, we shall derive the asymptotical stability in the mean square for nonlinear -Laplacian diffusion system (8).
It follows by (H1), (H3), and Lemma 3 that Similarly,
On the other hand,Similarly,
Next, we need to recall some facts derived by mathematical analysis. Assume that is continuous on variables and , and exists, utilizing the integral middle value theorem reaches where both and are differentiable.
Moreover, we can derive by employing (32) time and again
Now the standard Lyapunov functional theory derives that the null solution of the fuzzy system (8) is globally asymptotically stable.
Theorem 6 derives the following corollary.
Corollary 10. If there exists a positive definite matrix and two positive scalars , such that the following inequalities hold: then the null solution of the ordinary integrodifferential equations (9) is globally asymptotically stable.
Furthermore, if both diffusion behaviors and distributed delay are ignored, we derive from Corollary 10.
Corollary 11. If there exists a positive definite matrix and two positive scalars , such that the following inequalities hold: then the null solution of the following fuzzy system is globally asymptotically stable.
Discussion 1. In recent related literature ([27, 45–51]), some new conditions and methods were presented, and their results were very good. However, some of the methods and conditions are not applicable for system (8) with nonlinear -Laplacian diffusion. How to apply the new conditions and methods of [45–49] to our system (8) is an interesting problem.
4. Methods and Numerical Example
In this paper, Lyapunov functional method is employed to derive the stability criterion. In this process, the integral middle value theorem together with the derivation formula on integral upper limit functions plays the important roles.
Example 1. Consider the following Takagi-Sugeno -Laplace fuzzy T-S dynamic equations.
Fuzzy Rule 1. IF is , and is , THENFuzzy Rule 2. IF is , and is , THENwhere , and then Remark 1 gives Let , and then . Let ,