Delay-Dependent Asymptotic Stability of Cohen-Grossberg Models with Multiple Time-Varying Delays

Dynamical behavior of a class of Cohen-Grossberg models with multiple time-varying delays is studied in detail. Sufficient delay-dependent criteria to ensure local and global asymptotic stabilities of the equilibrium of this network are derived by constructing suitable Lyapunov functionals. The obtained conditions are shown to be less conservative and restrictive than those reported in the known literature. Some numerical examples are included to demonstrate our results.


Introduction
A large class of neural networks, which can function as stable content addressable memories or CAMs [1,2], had been proposed by Cohen and Grossberg.These Cohen-Grossberg networks were designed to include additive neural networks, later studied by Hopfield [3,4], and shunting neural networks.In the original analysis, Cohen and Grossberg assumed that the weight matrix was symmetric.Meanwhile, the activation functions are assumed to be continuous, differentiable, monotonically increasing, and bounded, such as the sigmoid-type function.Usually, such systems have been investigated under the assumption of asymmetric connection weight and nonmonotonic activation function.However, monotonicity and differentiability of activation functions come form the experimental results of brain sciences, moreover, they have very strong biological background.On the other hand, realistic modeling of many large neural networks with nonlocal interaction inevitably requires connection delays to be taken into account, since they naturally arise as a consequence of finite information transmission and processing speeds among the neurons.
It is also important to incorporate time delay into the model equations of the network such as delayed cellular neural network, which can be used to solve problems like the processing of moving images [5,6].Ye et al. [7] introduced discrete delays into the Cohen-Grossberg model.Furthermore, their global stability needed to satisfy the requirements that the connection should possess certain amount of symmetry and the discrete delays were sufficiently small.
For the delayed Hopfield networks [7][8][9][10][11][12][13][14], cellular neural networks [5,6], as well as BAM networks [15][16][17][18], some delay-independent criteria for the global asymptotic stability are established without assuming the monotonicity and the differentiability of the activation functions and also the symmetry of the connection.Wang and Zou [19] also studied the Cohen-Grossberg model with time delays.The global stability criteria of this type of neural networks were also obtained by constructing appropriate Lyapunov functionals, or Lyapunov functions combined with the Rezumikhin technique.All of these criteria are independent of the magnitudes of the delays, and therefore the delays are harmless in a network satisfying one of the criteria.Actually, the global exponential stability implies global asymptotic stability, and so the results leading to global exponential stability can provide relevant estimates on how fast such networks perform during realtime computations.Furthermore, Liao et al. [20,21] studied this problem.
Generally, the stability criteria for time-delay systems can be classified into two categories, namely delay-independent criteria and delay-dependent criteria, depending on whether they contain the delay argument as a parameter.There have been a number of significant developments in searching the stability criteria for systems with constant delays [4,6,7,9,11,12,[14][15][16][17].Only a few of them are for neural networks with distributed delays; see, for instance, [1-5, 8, 10, 18].To the best of the authors' knowledge, the delaydependent criteria in the case of the delayed Cohen-Grossberg model are little studied yet.In this paper, we will present some new local and global asymptotic stabilities of the equilibrium of Cohen-Grossberg models with mulitiple delays.Our results essentially show that the equilibrium of the network remains globally asymptotically stable when the time delays are small enough.In order to prove our results, we construct the suitable Lyapunov functionals.
In this paper, the amplification functions need to be continuous, positive, and bounded.However, the self-signal functions are not assumed to be differentiable, but only need to satisfy condition (H 2 ), as stated in the next section.At the same time, we do not confine ourselves to the symmetric connections.The rest of this paper is organized as follows.In Section 2, the Cohen-Grossberg neural network with time-varying delays and some preliminary analyses are given.By constructing Lyapunov functionals, some global exponential stability criteria for the network are presented in Section 3. Finally, numerical example is given to illustrate our results and some conclusions are drawn in Section 4.

Some preliminaries and network models
We consider Cohen-Grossberg neural networks with multiple time-varying delays, described by equations of theform X. Liao and S. Guo 3 ui where u i denotes the state variable associated with the ith neuron, the function a i represents an amplification function, and b i is an arbitrary function; however we will require that b i be sufficiently well behaved to keep the solutions of (2.1) bounded.The t (k) i j 's denote the interconnections which are associated with delay τ k (t), τ k (t) denotes the kth time delay for k = 0,1,2,...,K such that 0 System (2.1) is said to be globally stable if for any solution u(t), lim t→∞ u(t) exists.For the definitions of stability and asymptotic stability of an equilibrium of (2.1), refer to any of several standard texts (see, e.g., [22]).
In this paper, we assume that the Cohen-Grossberg neural networks (2.1) satisfy the following assumptions.
(H 1 ) The function a i is bounded, positive, and continuous.(H 2 ) The function b i is continuous, and there exist positive constants B i and B i , i = 0,1,2,...,n, such that (2.2) The initial condition for system (2.1) is given as follows: Proof.We only need to consider system (2.1).We know by (H 1 )-(H 4 ) that the terms whenever u i (t) ≤ −M for all i = 1,2,...,n.Since a i (u i (t)) is positive by (H 1 ), it can be concluded that for any solution u(t) of system (2.1), ui (t) < 0 whenever u i (t) ≥ M and ui (t) > 0 whenever u i (t) ≤ −M for all i = 1,2,...,n.We may assume that for the initial condition φ j (s), |φ j (s)| < M, otherwise we just pick a larger M. Thus we can conclude that u i (t) ≤ M for all t ≥ 0 and all i = 1,2,...,n.
It is also easy to show that (2.1) has always an equilibrium u * j , i = 1,2,...,n.That is, there exist u * j , i = 1,2,...,n, such that By using the strict monotonicity property of b i , there exist positive numbers b i > 0, Thus, In fact, let us consider the map P = (P 1 ,P 2 ,...,P n ) on the compact convex set Ω, where (2.9) It follows from (H 1 ) that P is a continuous map Ω into itself.Thus, it follows from Brouwer's fixed point theorem (see, e.g., [22]) that P has at least one fixed point where the positive constants N i , i = 1,2,...,n, satisfy (2.12) Thus, for sufficiently small η > 0 and sufficiently large T 0 > 0, such that for t ≥ T 0 , which together with (H 3 ) and (2.1) yield that for t ≥ T 0 , (2.15) Note that one can take η → 0 as t → +∞, we have where (2.17) By repeating the above procedure, we can obtain positive sequences {N i,k } such that Let N i denote the limits of {N i,k } as k → +∞, respectively.Then, we have (2. 19) This shows that Lemma 2.2 holds.
Lemma 2.4.Let f be a nonnegative function defined on R + such that f is integrable and uniformly continuous on R + .Then lim t→+∞ f (t) = 0.

Stability analysis
In this section, we will consider the stability of the equilibrium (u * 1 ,u * 2 ,...,u * n ) of system (2.1).
(H 5 ) There exist positive constants λ i , i = 1,2,...,n, such that By the process of the proof of Theorem 3.1, we can easily obtain the following.
(3.27) Hence, we can easily have the following.

Numerical example and conclusions
Generally, the delay-independent criteria are particularly restrictive and conservative for networks parameters.Moreover, it is reasonable to consider and apply these criteria first.If they are found inappropriate, the delay-dependent criteria will then be applied.To illustrate the results presented in Theorem 3.1 and Corollary 3.2, a simple example is given and a comparison of the results is given based on the results of literature [7] in the following.
We consider the following model system: ẋ1 (t) = − 4 + sin x 1 (t) 2x  We can easily find that the delay-independent conditions given in [7] are not applied and satisfied.This demonstrates that the delay-independent criteria are more conservative and restrictive than the delay-dependent criteria.
For system (4.1),we can obtain τ < 0.2828 from [7,Theorem 3.1].However, we can also obtain τ ≤ 0.8246 based on our results of Theorem 3.1.Numerical simulations have also been performed (see Figures 4.1,4.2,4.3 and 4.4).However, the problem of whether the delay superbound is optimal will be studied in a forthcoming paper.
In this paper, we have analyzed Cohen-Grossberg model with time delays in detail.The global asymptotic stability criteria for the equilibrium are derived based on the approach of Lyapunov functional.The obtained results are delay-dependent.Then, the X.Liao and S. Guo 15 delay-dependent criteria for local asymptotic stability criteria have also been obtained.Hence, our work has complemented and generalized that reported in [7].