Stochastic Dynamics of Nonautonomous Cohen-Grossberg Neural Networks

and Applied Analysis 3 see 35 . When designing an associative memory neural network, we should make convergence speed as high as possible to ensure the quick convergence of the network operation. Therefore, pth moment p ≥ 2 exponential stability and almost sure exponential stability are most useful concepts as they imply that the solutions will tend to the trivial solution exponentially fast. This motivates us to study pth moment exponential stability, and almost sure exponential stability for System 1.1 in this paper. The remainder of this paper is organized as follows. In Section 2, the basic assumptions and preliminaries are introduced. After establishing the criteria for the pth moment p ≥ 2 exponential stability and almost sure exponential stability for System 1.1 by using the Lyapunov functionmethod, Burkholder-Davids-Gundy inequality and Borel-Cantell’s theory in Section 3, an illustrative example and its simulations are given in Section 4. 2. Preliminaries Throughout this article, we let Ω,F, {Ft}t≥0, P be a complete probability space with a filtration {Ft}t≥0 satisfying the usual conditions i.e., it is right continuous and F0 contains all P -null sets . Let C C −∞, 0 , R be the Banach space of continuous functions which map into R with the topology of uniform convergence. For any x t x1 t , . . . , xn t T ∈ R, we define ‖x t ‖ ‖x t ‖p ∑n i 1 |xi t |p , 1 ≤ p < ∞ . The initial conditions for system 1.1 are x s φ s ,−τ ≤ s ≤ 0, φ ∈ LpF0 −τ, 0 , R ; here LpF0 −τ, 0 , R is R-valued stochastic process φ s ,−τ ≤ s ≤ 0, φ s is F0-measurable, ∫0 −τ E|φ s |pds < ∞. For the sake of convenience, throughout this paper, we assure fj 0 gj 0 σij 0 0, which implies that system 1.1 admits an equilibrium solution x t ≡ 0. If V ∈ C2,1 −τ,∞ × R;R , according to the Itô formula, define an operator LV associated with 1.2 as LV t, x Vt Vx{−H x t C x t −A t F x t − B t G xτ t } 1 2 trace [ σVxxσ ] , 2.1 where Vt ∂V t, x /∂t, Vx ∂V t, x /∂x1, . . . , ∂V t, x /∂xn , and Vxx ∂2V t, x / ∂xi∂xj n×n. To establish the main results of the model given in 1.1 , some of the standing assumptions are formulated as follows: H1 there exist positive constants hi, hi, such that 0 < hi ≤ hi x ≤ hi < ∞, ∀x ∈ R, i 1, 2, . . . , n; 2.2 H2 for each i 1, 2, . . . , n, there exist positive functions αi t > 0, such that xi t ci xi t ≥ αi t x2 i t ; 2.3 4 Abstract and Applied Analysis H3 there exist positive constants βj , γj , i, j 1, 2, . . . , n, such that ∣fj u − fj v ∣∣ ≤ βj |u − v|, ∣gj u − gj v ∣∣ ≤ γj |u − v|; 2.4 H4 each σij x satisfies the Lipschitz condition, and there exist positive constants μi, i 1, 2, . . . , n, such that trace { σ x σ x } ≤ n ∑ i 1 μix 2 i . 2.5 Remark 2.1. The activation functions are typically assumed to be continuous, differentiable, and monotonically increasing, such as the functions of sigmoid type. These restrictive conditions are no longer needed in this paper. Instead, only the Lipschitz condition is imposed in Assumption H3 . Note that the type of activation functions in H3 have already been used in numerous papers, see 5, 10 and references therein. Remark 2.2. We remark here that non-autonomous conditions H2 – H4 replace the usual autonomous conditions which is more useful for practical purpose; please refer to 4, 13 and references therein. Remark 2.3. The delay functions τj t considered in this paper only needed to be bounded; they can be time-varying, nondifferentiable functions. This generalized some recently published results in 4, 13, 26–29 . Different from the models considered in 4, 13, 29 , in this paper, we have removed the following condition: H0 For each j 1, 2, . . . , n, τj t is a differentiable function, namely, there exists ξ such that τ̇j t ≤ ξ < 1. 2.6 Definition 2.4 see 35 . The trivial solution of 1.1 is said to be pth moment exponential stability if there is a pair of positive constants λ and C such that E‖x t, t0, x0 ‖p ≤ C‖x0‖pe−λ t−t0 , on t ≥ t0, ∀x0 ∈ R, 2.7 where p ≥ 2 is a constant; when p 2, it is usually said to be exponential stability in mean square. Definition 2.5 see 35 . The trivial solution of 1.1 is said to be almost sure exponential stability if for almost all sample paths of the solution x t , we have lim sup t→∞ 1 t log‖x t ‖ < 0. 2.8 Abstract and Applied Analysis 5 Lemma 2.6 35 Burkholder-Davids-Gundy inequality . There exists a universal constant Kp for any 0 < p < ∞ such that for every continuous local martingale M vanishing at zero and any stopping time η,and Applied Analysis 5 Lemma 2.6 35 Burkholder-Davids-Gundy inequality . There exists a universal constant Kp for any 0 < p < ∞ such that for every continuous local martingale M vanishing at zero and any stopping time η, E ( sup 0≤s≤η |Ms| ) ≤ KpE ( 〈M,M〉η )p/2 , 2.9 where 〈M,M〉η is the cross-variation ofM. In particular, one may haveKp 32/p p/2 if 0 < P < 2 and K2 4 if p 2; although they may not be optimal, for example, one could haveK1 4 √ 2. Lemma 2.7 35 Chebyshev’s inequality . P{ω : |X ω | ≥ c} ≤ c−pE|X|p 2.10 if c > 0, p > 0, X ∈ L. Lemma 2.8 36 Borel-Cantell’s lemma . Let {An, n ≥ 1} be a sequence of events in some probability space, then i if ∑∞ n 1 P An < ∞, then P An, i.o. 0; ii moreover, if {An, n ≥ 1} are independent of each other, then ∑∞ n 1 P An ∞ implies P An, i.o. 1, 2.11 where {An, i.o.} denotes occurring infinitely often within {An, n ≥ 1}, that is, {An, i.o.} ∩∞ k 1∪n kAn. “i.o.” is the abbreviation of “infinitely often”. 3. Main Results Theorem 3.1. Under the assumptions (H1)–(H4), if there are a positive diagonal matrix M diag m1, . . . , mn and two constants 0 < N2, 0 ≤ μ < 1, such that 0 < N2 ≤ N2 t ≤ μN1 t , for t ≥ t0, 3.1