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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 396903, 21 pages
LMI Approach to Exponential Stability and Almost Sure Exponential Stability for Stochastic Fuzzy Markovian-Jumping Cohen-Grossberg Neural Networks with Nonlinear -Laplace Diffusion
1Institution of Mathematics, Yibin University, Yibin, Sichuan 644007, China
2School of Science Mathematics, University of Electronic Science and Technology of China, Chengdu 610054, China
3College of Mathematics and Software Science, Sichuan Normal University, Chengdu 610066, China
Received 3 February 2013; Accepted 23 March 2013
Academic Editor: Qiankun Song
Copyright © 2013 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.
The robust exponential stability of delayed fuzzy Markovian-jumping Cohen-Grossberg neural networks (CGNNs) with nonlinear -Laplace diffusion is studied. Fuzzy mathematical model brings a great difficulty in setting up LMI criteria for the stability, and stochastic functional differential equations model with nonlinear diffusion makes it harder. To study the stability of fuzzy CGNNs with diffusion, we have to construct a Lyapunov-Krasovskii functional in non-matrix form. But stochastic mathematical formulae are always described in matrix forms. By way of some variational methods in , Itô formula, Dynkin formula, the semi-martingale convergence theorem, Schur Complement Theorem, and LMI technique, the LMI-based criteria on the robust exponential stability and almost sure exponential robust stability are finally obtained, the feasibility of which can efficiently be computed and confirmed by computer MatLab LMI toolbox. It is worth mentioning that even corollaries of the main results of this paper improve some recent related existing results. Moreover, some numerical examples are presented to illustrate the effectiveness and less conservatism of the proposed method due to the significant improvement in the allowable upper bounds of time delays.
It is well known that in 1983, Cohen-Grossberg  proposed originally the Cohen-Grossberg neural networks (CGNNs). Since then the CGNNs have found their extensive applications in pattern recognition, image and signal processing, quadratic optimization, and artificial intelligence [2–6]. However, these successful applications are greatly dependent on the stability of the neural networks, which is also a crucial feature in the design of the neural networks. In practice, time delays always occur unavoidably due to the finite switching speed of neurons and amplifiers [2–8], which may cause undesirable dynamic network behaviors such as oscillation and instability. Besides delay effects, stochastic effects also exist in real systems. In fact, many dynamical systems have variable structures subject to stochastic abrupt changes, which may result from abrupt phenomena such as sudden environment changes, repairs of the components, changes in the interconnections of subsystems, and stochastic failures. (see  and references therein). The stability problems for stochastic systems, in particular the Itô-type stochastic systems, become important in both continuous-time case and discrete-time case . In addition, neural networks with Markovian jumping parameters have been extensively studied due to the fact that systems with Markovian jumping parameters are useful in modeling abrupt phenomena, such as random failures, operating in different points of a nonlinear plant, and changing in the interconnections of subsystems [11–15].
Remark 1. Deterministic system is only the simple simulation for the real system. Indeed, to model a system realistically, a degree of randomness should be incorporated into the model due to various inevitable stochastic factors. For example, in real nervous systems, synaptic transmission is a noisy process brought on by random fluctuations from the release of neurotransmitters and other probabilistic causes. It is showed that the above-mentioned stochastic factors likewise cause undesirable dynamic network behaviors and possibly lead to instability. So it is of significant importance to consider stochastic effects for neural networks. In recent years, the stability of stochastic neural networks has become a hot study topic [3, 16–21].
On the other hand, diffusion phenomena cannot be unavoidable in real world. Usually diffusion phenomena are simply simulated by linear Laplace diffusion in much of the previous literature [2, 22–24]. However, diffusion behavior is so complicated that the nonlinear reaction-diffusion models were considered in several papers [3, 25–28]. Very recently, the nonlinear -Laplace diffusion () is applied to the simulation of some diffusion behaviors . But almost all of the above mentioned works were focused on the traditional neural networks models without fuzzy logic. In the factual operations, we always encounter some inconveniences such as the complicity, the uncertainty and vagueness. As far as we know, vagueness is always opposite to exactness. To a certain degree, vagueness cannot be avoided in the human way of regarding the world. Actually, vague notations are often applied to explain some extensive detailed descriptions. As a result, fuzzy theory is regarded as the most suitable setting to taking vagueness and uncertainty into consideration. In 1996, Yang and his coauthor  originally introduced the fuzzy cellular neural networks integrating fuzzy logic into the structure of traditional neural networks and maintaining local connectedness among cells. Moreover, the fuzzy neural network is viewed as a very useful paradigm for image processing problems since it has fuzzy logic between its template input and/or output besides the sum of product operation. In addition, the fuzzy neural network is a cornerstone in image processing and pattern recognition. And hence, investigations on the stability of fuzzy neural networks have attracted a great deal of attention [30–37]. Note that stochastic stability for the delayed -Laplace diffusion stochastic fuzzy CGNNs have never been considered. Besides, the stochastic exponential stability always remains the key factor of concern owing to its importance in designing a neural network, and such a situation motivates our present study. Moreover, the robustness result is also a matter of urgent concern [10, 38–46], for it is difficult to achieve the exact parameters in practical implementations. So in this paper, we will investigate the stochastic global exponential robust stability criteria for the nonlinear reaction-diffusion stochastic fuzzy Markovian-jumping CGNNs by means of linear matrix inequalities (LMIs) approach.
Both the non-linear -Laplace diffusion and fuzzy mathematical model bring a great difficulty in setting up LMI criteria for the stability, and stochastic functional differential equations model with nonlinear diffusion makes it harder. To study the stability of fuzzy CGNNs with diffusion, we have to construct a Lyapunov-Krasovskii functional in non-matrix form (see, e.g., ). But stochastic mathematical formulae are always described in matrix forms. Note that there is no stability criteria for fuzzy CGNNs with -Laplace diffusion, let alone Markovian-jumping stochastic fuzzy CGNNs with -Laplace diffusion. Only the exponential stability of Itô-type stochastic CGNNs with -Laplace diffusion was studied by one literature  in 2012. Recently, Ahn use the passivity approach to derive a learning law to guarantee that Takagi-Sugeno fuzzy delayed neural networks are passive and asymptotically stable (see, e.g., [47, 48] and related literature [49–57]). Especially, LMI optimization approach for switched neural networks (see, e.g., ) may bring some new edificatory to our studying the stability criteria of Markovian jumping CGNNs. Muralisankar, Gopalakrishnan, Balasubramaniam, and Vembarasan investigated the LMI-based robust stability for Takagi-Sugeno fuzzy neural networks [36, 38–41]. Mathiyalagan et al. studied robust passivity criteria and exponential stability criteria for stochastic fuzzy systems [10, 37, 42–46]. Motivated by some recent related works ([9, 10, 36–57], and so on), particularly, Zhu and Li , Zhang et al. , Pan and Zhong , we are to investigate the exponential stability and robust stability of Itô-type stochastic Markovian jumping fuzzy CGNNs with -Laplace diffusion. By way of some variational methods in (Lemma 6), Itô formula, Dynkin formula, the semi-martingale convergence theorem, Schur Complement Theorem, and LMI technique, the LMI-based criteria on the (robust) exponential stability and almost sure exponential (robust) stability are finally obtained, the feasibility of which can efficiently be computed and confirmed by computer matlab LMI toolbox. When , or ignoring some fuzzy or stochastic effects, the simplified system may be investigated by existing literature (see, e.g., [2–4, 58]). Another purpose of this paper is to verify that some corollaries of our main results improve some existing results in the allowable upper bounds of time delays, which may be illustrated by numerical examples (see, e.g., Examples 30 and 36).
The rest of this paper is organized as follows. In Section 2, the new -Laplace diffusion fuzzy CGNNs models are formulated, and some preliminaries are given. In Section 3, new LMIs are established to guarantee the stochastic global exponential stability and almost sure exponential stability of the above-mentioned CGNNs. Particularly in Section 4, the robust exponential stability criteria are given. In Section 5, Examples 28, 30, 32, 35, 36, and 38 are presented to illustrate that the proposed methods improve significantly the allowable upper bounds of delays over some existing results ([4, Theorem 1], [4, Theorem 3], [58, Theorem 3.1], [58, Theorem 3.2]). Finally, some conclusions are presented in Section 6.
2. Model Description and Preliminaries
In 2012, Zhu and Li  consider the following stochastic fuzzy Cohen-Grossberg neural networks: where each is scalar standard Brownian motion defined on a complete probability space with a natural filtration . The noise perturbation is a Borel measurable function. and denote the fuzzy AND and OR operation, respectively. Under several inequalities conditions and the following five similar assumptions on System (1), some exponential stability results are obtained in . Of course, in this paper, we may present the following conditions which are more flexible than those of .(A1) There exists a positive definite diagonal matrix such that for all , .(A2) There exist positive definite diagonal matrix such that (A3) For any given , is locally Lipschitz continuous, and there exists a constant such that for all at which is differentiable; is locally Lipschitz continuous, and there exists a constant such that at which is differentiable.(A4) There exist nonnegative matrices and such that where , .(A5), , .
Remark 2. The condition (A3) is different from that of some existing literature (e.g., [2–4]). In those previous literature, and are always assumed to be globally Lipschitz continuous. Here, we relax this assumption, for and are only the local Lipschitz continuous functions. From Rademacher's theorem , a locally Lipschitz continuous function is differentiable almost everywhere. Let be the set of those points where is differentiable, then is the Jacobian of at and the set is dense in . The generalized Jacobian of a locally Lipschitz continuous function is a set of matrices defined by
where denotes the convex hull of a set.
The condition (A5) guarantees zero-solution is an equilibrium of stochastic fuzzy system (1). Throughout this paper, we always assume that all assumptions (A1)–(A5) hold. In addition, we assume that and are symmetric matrices in consideration of LMI-based criteria presented in this paper.
Besides delays, stochastic effects, the complexity, the vagueness and diffusion behaviors always occur in real nervous systems. So in this paper, we are to consider the following delays stochastic fuzzy Markovian-jumping Cohen-Grossberg neural networks with nonlinear -Laplace diffusion : with boundary conditionwhere is a given scalar, is a bounded domain with a smooth boundary of class by , , , and is the state variable of the th neuron and the th neuron at time and in space variable . The smooth nonnegative functions are diffusion operators. Time delay . represents an amplification function, and is an appropriately behavior function. , are neuron activation functions of the th unit at time and in space variable . is a right-continuous Markov process on the probability space which takes values in the finite space with generator given by where is transition probability rate from to and , and . In mode , we denote , , and , which imply the connection strengths of the ith neuron on the th neuron, respectively.
The boundary condition (6a) is called Dirichlet boundary condition if , and Neumann boundary condition if , where , denotes the outward normal derivative on . It is well known that the stability of neural networks with Neumann boundary condition has been widely studied. The Dirichlet boundary conditions describe the situation where the space is totally surrounded by a region in which the states of the neuron equal zero on the boundary. And the stability analysis of delayed reaction-diffusion neural networks with the Dirichlet boundary conditions is very important in theories and applications, and also has attracted much attention [2, 3, 29, 58]. So in this paper, we consider the CGNNs under Neumann boundary condition and Dirichlet boundary condition, respectively.
If the complexity and the vagueness of CGNNs are ignored, the stochastic fuzzy system (6) is simplified to the following stochastic system: where matrices , . In 2012, Wang et al.  studied the stability of System (8) without Markovian-jumping.
Finally, we consider the global robust exponential stability for the following uncertain fuzzy CGNNs with -Laplace diffusion:
For any mode , we denote , , , by , , , , and matrices , , , . Assume The , , , and are parametric uncertainties, satisfying where is an unknown matrix with , and , , , are known real constant matrices for all .
Throughout this paper, we denote matrices , , , , , , . For the sake of simplicity, let , and , . Matrix satisfies for all , , . Denote with . And denotes the Hadamard product of matrix and (see,  or ).
For convenience’s sake, we need introduce some standard notations.(i) The space of real Lebesgue measurable functions of , it is a Banach space for the 2-norm with , where is Euclid norm.(ii): The family of all -measurable -value random variable such that , where stands for the mathematical expectation operator with respect to the given probability measure .(iii) (<0): A positive (negative) definite matrix, that is, (<0) for any .(iv) : A semi-positive (semi-negative) definite matrix, that is, for any .(v): This means is a semi-positive (semi-negative) definite matrix.(vi): This means is a positive (negative) definite matrix.(vii) denotes the largest and smallest eigenvalue of matrix , respectively.(viii)Denote for any matrix ; for any .(ix): Identity matrix with compatible dimension.(x)The symmetric terms in a symmetric matrix are denoted by .(xi)The Sobolev space (see  for detail). Particularly in the case of , then .(xii)Denote by the lowest positive eigenvalue of the boundary value problem
Let be the state trajectory from the initial condition , on in . Below, we always assume is a solution of System (6).
Definition 4. For any given scalar , the null solution of system (6) is said to be stochastically globally exponentially stable in the mean square if for every initial condition , , there exist scalars and such that for any solution ,
Definition 5. The null solution of System (6) is said to be almost sure exponentially stable if for every , there exists a positive scalar such that the following inequality holds:
Lemma 8 (nonnegative semi-martingale convergence theorem ). Let and be two continuous adapted increasing processes on with . Let be a real-valued continuous local martingale with . Let be a nonnegative -measurable random variable with . Define
for . If is nonnegative, then
where . means . In particular, if ., then for almost all , and , that is, both and converge to finite random variables.
Lemma 9 (see ). Let be locally Lipschitz continuous. For any given , there exists an element in the union such that
where denotes the segment connecting and .
Lemma 10 (see ). Let be any given scalar, and and be matrices with appropriate dimensions. If , then one has
3. Main Results
Theorem 11. Assume that . In addition, there exist a sequence of positive scalars and positive definite diagonal matrices and such that the following LMI conditions hold:
where matrices , , , and
then the null solution of Markovian jumping stochastic fuzzy system (6) is stochastically exponentially stable in the mean square.
Proof. Consider the Lyapunov-Krasovskii functional:
where is a solution for stochastic fuzzy system (6). Sometimes we may denote by , by , and , by for simplicity.
Let be the weak infinitesimal operator. Then it follows by Lemma 6 that
Moreover, we get by A4 and A5
From A3 and Lemma 9, we know
where , and .
So it follows by A1–A5 that
Remark 12. In (28), we employ a new method, which is different from that of [4, (3)]. Therefore, our LMI conditions in Theorem 11 may be more feasible and effective than [4, Theorem 1] to some extent, which may be illustrated by a numerical example below (see, Example 30).
Denote . Then we get by (21)
Then we can obtain by the Dynkin formula
Hence, we have
On the other hand,
Combining the above two inequalities, we obtain
Therefore, we can see it by Definition 4 that the null solution of stochastic fuzzy system (6) is globally stochastically exponentially stable in the mean square.
Corollary 13. If there exist a positive scalar and positive definite diagonal matrices and such that the following LMI conditions hold:
where matrices ,,,, and
then the null solution of stochastic fuzzy system (1) is stochastically exponentially stable in the mean square.
Remark 14. It is obvious from Remark 12 that our Corollary 13 is more feasible and effective than [4, Theorem 1]. In addition, the LMI-based criterion of Corollary 13 has its practical value in real work, for it is available to computer matlab calculation.
Corollary 15. Assume that . In addition, there exist a sequence of positive scalars and positive definite diagonal matrices and such that the following LMI conditions hold:
where then the null solution of Markovian jumping stochastic system (8) is stochastically exponentially stable in the mean square.
Particularly for the case of , we get from the Poincaré inequality (see, e.g., [58, Lemma 2.4]) that
Denote . Then Lemma 6 derives that
where , and is a positive scalar, satisfying Moreover, one can conclude the following Corollary from (40) and the proof of Theorem 11.
Corollary 16. Assume that . In addition, there exist a sequence of positive scalars and positive definite diagonal matrices and such that the following LMI conditions hold:
where satisfies (38), then the null solution of Markovian jumping stochastic system (8) with is stochastically exponentially stable in the mean square.
Below, we denote for convenience's sake.
Theorem 18. Assume . The null solution of Markovian jumping stochastic fuzzy system (6) is almost sure exponentially stable if there exist positive scalars , , and positive definite matrices such that
Proof. Consider the Lyapunov-Krasovskii functional:
By applying Itô formula (see, e.g., [3, (2.7)]) and Lemma 6, we can get
where , , and .
Similarly as (26)–(29), we can derive by A1–A5 and Lemma 9
On the other hand,
Denote for convenience’s sake. Then we have
Combining (47) and (49) results in
which together with (43) implies
Remark 19. The methods employed in (38)–(50) are different from ones in the proof of [4, Theorem 3] so that our efficient LMI criterion can be constructed. In large numerical calculations, LMI-based criterion in Theorem 18 is more effective than the complicated condition (8) in [4, Theorem 3]. To some extent, Theorem 18 is more effective than [58, Theorem 3.1] to some extent if fuzzy system (6) is simplified to system (8) without Markovian jumping (see, e.g., Example 38).
It is obvious that the right-hand side of (51) is a nonnegative semi-martingale. And hence the semi-martingale convergence theorem derives
Then we can conclude from (52)
Corollary 20. The null solution of stochastic fuzzy system (1) is almost sure exponentially stable if there exist positive scalars , , and positive definite diagonal matrices such that
Corollary 22. Assume . The null solution of Markovian jumping stochastic system (8) is almost sure exponentially stable if there exist positive scalars , , and positive definite matrices , such that
Corollary 23. Assumed . The null solution of Markovian jumping stochastic system (8) is almost sure exponentially stable if there exist positive scalars , , , and positive definite matrices , such that
where satisfies (58).
Remark 24. As pointed out in Remark 19, Corollary 23 is obviously more effective than [58, Theorem 3.1] if Markovian jumping stochastic system (8) is simplified to a stochastic system without Markovian juamping.
4. The Robust Exponential Stability Criteria
Theorem 25. Assume that . In addition, there exist a sequence of positive scalars and positive definite diagonal matrices and such that for all ,
where matrices , , , , and then the null solution of Markovian jumping stochastic fuzzy system (9) is stochastically exponentially robust stable in the mean square.