LMI Approach to Stability Analysis of Cohen-Grossberg Neural Networks with p-Laplace Diffusion

The nonlinear p-Laplace diffusion p > 1 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 p 2, p-Laplace diffusion is just the conventional Laplace diffusion in many previous literatures. And it is worth mentioning that even if p 2, 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.


Introduction and Preparation
It is well known that Cohen-Grossbeg neural network CGNN was proposed by Cohen and Grossberg 1 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 19 .So in this paper, we investigate the stability of the following stochastic CGNN with nonlinear p-Laplace diffusion p > 1 : x is more natural than that of 5 , which will result in some difference in methods.
Here, M is a diagonal matrix, Ω ∈ R m is a bounded compact set with smooth boundary, u u 1 , u 2 , . . ., u n T ∈ R n , and w t w 1 t , w 2 t , . . ., w n t T is a n-dimensional Brownian motion defined on a complete probability space Ω, F, P with the natural filtration {F t } t≥0 generated by the process {w s : 0 ≤ s ≤ t}.We associate Ω with the canonical space generated by all {w i t } and denoted by F the associated σ-algebra generated by w t with the probability measure p.A u t, x presents an amplification function, B u t, x is an appropriately behavior function, and f and g denote the activation function.τ t 0 ≤ τ t ≤ τ corresponds to the transmission delays at time, and t k is called the impulsive moment with 0 < t Particularly, ∇ p u ∇u for the case of p 2.
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 p-Laplace diffusion is considered in System 1.1 .
Assume, in addition, the following.H1 A u t, x is a bounded, positive, and continuous diagonal matrix, that is, there exist two positive diagonal matrices A and A such that 0 < A ≤ A u t, x ≤ A.
H3 There exist two positive diagonal matrices H4 The null solution is the equilibrium point of system 1.1 , that is, the following conditions hold: where the symmetrical matrix Q > 0. For convenience's sake, we introduce some standard notations 2 < ∞, where E{•} stands for the mathematical expectation operator with respect to the given probability measure p.
iii Q q ij n×n > 0 <0 : a positive negative definite symmetrical matrix, that is, y T Qy > 0 <0 for any 0 / y ∈ R n .iv Q q ij n×n ≥ 0 ≤0 : a semipositive semi-negative definite symmetrical matrix, that is, y T Qy ≥ 0 ≤0 for any y Definition 1.3.The null solution of impulsive system 2.2 is globally stochastically exponentially stable in the mean square if for every ϕ ∈ L 2 F 0 −τ, 0 × Ω; R n , there exists scalars β > 0 and γ > 0 such that 1.4 Lemma 1.4 see 11 .Let U, P be any matrices, ε > 0 is a positive number and matrix H H T > 0, then where matrix S t and symmetrical matrices Q t and R t depend on t, is equivalent to any one of the following conditions: Lemma 1.6 see 21 .Consider the following differential inequality: 7 sup t−τ≤s≤t v s and v t is continuous except t k , k 1, 2, . .., where it has jump discontinuities.The sequence t k satisfies 0 t δτ, where δ > 1, and there exist constants γ > 0, M > 0 such that where ρ i max{1, a i b i e λτ }, λ > 0 is the unique solution of equation λ a − be λτ then 1.9 In addition, if θ sup k∈Z {1, a k b k e λτ }, then

2.2
Construct the Lyapunov functional as follows: where u T s, x P 2 u s, x ds dx.

2.5
And then we have Next, we use the method similar as that of 22 .Since u t, x is the solution of system, and

2.12
Next, we have

2.13
Now the conditions C1 -C3 and Lemma 1.6 deduce  where , and the corresponding matrices We might as well assume that t 0 0, t k − t k−1 0.525, τ t 0.65, τ t ≤ μ 0.99 for all t ≥ t 0 , and

Conclusions
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 p-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 p 2 and has more computational efficiency due to the LMI approach even if p 2 Remark 2.2 .

1 Figure 1 :Figure 2 :
Figure 1: Computer simulation of the state u 1 t, x .

Figure 3 :
Figure 3: Computer simulation of the state u 2 t, x .

Figure 4 :
Figure 4: Sectional curve of the state variable u 2 t, x .
n×m is Hadamard product of matrix D and ∇ p u 20 .
R for all t, we can get by It o formula Ω u T s − τ s , x G 2 μ − 1 P 2 u s − τ s , x dx ds, t ∈ t k , t k 1 .