LMI Approach to Stability Analysis of Cohen-Grossberg Neural Networks with -Laplace Diffusion
The nonlinear -Laplace diffusion () was considered in the Cohen-Grossberg neural network (CGNN), and a new linear matrix inequalities (LMI) criterion is obtained, which ensures the equilibrium of CGNN is stochastically exponentially stable. Note that, if , -Laplace diffusion is just the conventional Laplace diffusion in many previous literatures. And it is worth mentioning that even if , the new criterion improves some recent ones due to computational efficiency. In addition, the resulting criterion has advantages over some previous ones in that both the impulsive assumption and diffusion simulation are more natural than those of some recent literatures.
1. Introduction and Preparation
It is well known that Cohen-Grossbeg neural network (CGNN) was proposed by Cohen and Grossberg  in 1983. Since then there have been a lot of interested results obtained in many literatures (see [2–9]) due to its general applications, such as pattern recognition, image and signal processing, optimization automatic control, and artificial intelligence. Usually, there exist the impulsive effect and time-varying delays phenomenon in various neural networks [3, 5–7, 10–14]. Besides, diffusion effects cannot be avoided in the neural networks when electrons are moving in asymmetric electromagnetic fields [15–18]. However, diffusion disturbance was always simulated simply by linear Laplace diffusion [15–18]. Few papers involved the nonlinear reaction-diffusion . So in this paper, we investigate the stability of the following stochastic CGNN with nonlinear -Laplace diffusion (): where is a bounded subset in with smooth boundary , and denotes the outward normal derivative on .
Remark 1.1. If , system (1.1) was studied by  though there is a little difference between Dirichlet boundary condition and Neumann boundary condition. However, our impulsive assumption is more natural than that of , which will result in some difference in methods.
Here, is a diagonal matrix, is a bounded compact set with smooth boundary, , and is a -dimensional Brownian motion defined on a complete probability space with the natural filtration generated by the process . We associate with the canonical space generated by all and denoted by the associated -algebra generated by with the probability measure . presents an amplification function, is an appropriately behavior function, and and denote the activation function. corresponds to the transmission delays at time, and is called the impulsive moment with and . We always assume . , , is Hadamard product of matrix and . Here, the diffusion parameters matrix is denoted simply as . Let , and matrix , and we denote . Particularly, for the case of .
Remark 1.2. Diffusion effects always occur in the neural networks when electrons are moving in asymmetric electromagnetic fields [15–18], and diffusion behavior is so complicated that it cannot always be simulated by linear Laplace diffusion. So in this paper, the nonlinear -Laplace diffusion is considered in System (1.1).
Assume, in addition, the following.(H1) is a bounded, positive, and continuous diagonal matrix, that is, there exist two positive diagonal matrices and such that .(H2) such that there exists a positive diagonal matrix satisfying for all .(H3) There exist two positive diagonal matrices and such that (H4) The null solution is the equilibrium point of system (1.1), that is, the following conditions hold: where the symmetrical matrix .
For convenience's sake, we 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 symmetrical matrix, that is, (<0) for any .(iv) (≤0): a semipositive (semi-negative) definite symmetrical matrix, that is, (≤0) for any .(v): this means is a semi-positive (semi-negative) symmetrical definite matrix.(vi): this means is a positive (negative) symmetrical definite matrix.(vii) denotes the largest and smallest eigenvalue of symmetrical matrix , respectively.(viii): identity matrix with compatible dimension.(ix)Denote for any matrix ; for any .
Let denote the state trajectory from the initial data on in .
Definition 1.3. The null solution of impulsive system (2.2) is globally stochastically exponentially stable in the mean square if for every , there exists scalars and such that
Lemma 1.4 (see ). Let be any matrices, is a positive number and matrix , then
Lemma 1.5 (Schur complement ). The LMI where matrix and symmetrical matrices and depend on , is equivalent to any one of the following conditions: (L1); (L2).
Lemma 1.6 (see ). Consider the following differential inequality: where and is continuous except , where it has jump discontinuities. The sequence satisfies , and . Suppose that(1); (2), where , and there exist constants such that where , is the unique solution of equation then In addition, if , then
2. Main Results
Theorem 2.1. If assumptions (H1)–(H4) hold, in addition, the following conditions are satisfied:(C1) there exists diagonal matrices and such that (C2), where , where for all ;(C3) there exists a constant such that , and , where is the unique solution of the equation , and , , then the null solution of system (1.1) is stochastically exponentially stable with convergence rate .
Proof. First, we can get by Guass formula (see [20, Lemma 2.3])
Construct the Lyapunov functional as follows: where
Then And then we have
Next, we use the method similar as that of . Since is the solution of system, and for all , we can get by formula
Then we have
Thus, for small enough , we have and then Since
Then, Next, we have
Now the conditions (C1)–(C3) and Lemma 1.6 deduce or which together with Definition 1.3 implies the accomplishment of the proof.
Remark 2.2. The nonlinear -Laplace diffusion brings a great difficulties in judging the stability. However, even if , Theorem 2.1 has more computational efficiency than [15, Theorem 3.1] due to LMI criterion.
Consider the following impulsive CGNN: where , and the corresponding matrices We might as well assume that for all , and By way of MATLAB LMI Control Toolbox, we can solve the LMI condition in (C1) and get Next, we will prove that such and make (C2) and (C3) hold. Indeed, by computing directly, we can obtain , and then (C2) holds. Moreover, we might as well assume , and then we have , thus (C3) is satisfied. Now from Theorem 2.1 we can compute the convergence .
In this paper, we investigate the influence of impulse, time-delays and diffusion behaviors on the stability of stochastic Cohen-Grossberg neural network (CGNN). The LMI conditions of stochastic exponential stability of impulsive CGNN with -Laplace reaction-diffusion terms was given, and an illustrate example was also given to show the effectiveness of the obtained result. Besides, the result obtained in this paper is also valid to the Laplace reaction-diffusion (in the case of ) and has more computational efficiency due to the LMI approach even if (Remark 2.2).
This work was supported by the National Basic Research Program of China (2010CB732501), by Scientific Research Fund of Science Technology Department of Sichuan Province 2011JYZ010, and by Scientific Research Fund of Sichuan Provincial Education Department (11ZA172, 12ZB349).
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