Global exponential stability analysis of Cohen–Grossberg neural network with delays

doi:10.1016/j.cnsns.2006.09.004

Communications in Nonlinear Science and Numerical Simulation

Volume 13, Issue 5, July 2008, Pages 847-856

https://doi.org/10.1016/j.cnsns.2006.09.004 Get rights and content

Abstract

This brief presents new sufficient conditions of globally exponential stability of Cohen–Grossberg neural networks with time delays. The results are also compared with previously reported results in the literature, implying that the results obtained in this paper provide one more set of criteria for determining the global exponential stability of Cohen–Grossberg neural networks with time delays.

Introduction

Recently, the dynamical behavior of different classes of neural networks such as Cohen–Grossberg neural networks, Hopfield-type of neural networks and cellular neural networks have been attracted increasing attention due to their applicability in solving some optimization problems and pattern recognition problems. When neural networks are used to solve optimization problems, it is essential to determine the existence of a unique equilibrium point and its stability properties. As pointed out in [1], although neural networks are proved to be implemented by electronic circuits, the finite switching speed of amplifiers and the transmission delays during the communication between neurons will affect the dynamical behavior of neural networks. Therefore, determining the affects of the time delays on the equilibrium and stability properties of neural networks are of prime importance. Many researchers have studied the equilibria and stability properties of neural networks with delays and presented various sufficient conditions for the uniqueness and global exponential stability of the equilibrium point of neural networks with delays (see, for example, [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20]). The authors of [3], [8] analyzed the global asymptotic stability of Cohen–Grossberg neural networks with time-varying delays. In present letter, we focus on global exponential stability of Cohen–Grossberg neural networks with time-varying delays. By constructing a suitable Lyapunov functional, we obtain some new criteria for global exponential stability of Cohen–Grossberg neural networks with variable but bounded delays. Comparisons with some previous results are also made, it is shown that our results generalize some existing conditions in the literature.

The organization of this letter is as follows. In Section 2, we introduce the model of neural networks with delays and some notations and assumptions. A new criterion on the globally exponential stability of equilibrium point for neural networks with delays is given in Section 3. In Section 4, we shall present some remarks and examples to compare our criterion with the previous results. Some conclusions are given in Section 5.

Section snippets

Cohen–Grossberg neural network model and some basic concepts

Consider a neural networks with delays described by the state equations $\dot{x} (t) = D (x (t)) (- C (x (t)) + Af (x (t)) + Bf (x (t - τ (t))) + u),$ where $\begin{matrix} x (t) = (x_{1} (t), x_{2} (t), \dots, x_{n} (t))^{T}, \\ D (x (t)) = diag (d_{1} (x_{1} (t)), d_{2} (x_{2} (t)), \dots, d_{n} (x_{n} (t))), \\ C (x (t)) = (c_{1} (x_{1} (t)), c_{2} (x_{2} (t)), \dots, c_{n} (x_{n} (t)))^{T}, \\ A = (a_{ij})_{n \times n}, B = (b_{ij})_{n \times n}, \\ f (x (t)) = (f_{1} (x_{1} (t)), f_{2} (x_{2} (t)), \dots, f_{n} (x_{n} (t)))^{T}, \\ f (x (t - τ (t))) = (f_{1} (x_{1} (t - τ_{1} (t))), f_{2} (x_{2} (t - τ_{2} (t))), \dots, f_{n} (x_{n} (t - τ_{n} (t))))^{T}, \\ u = (u_{1}, u_{2}, \dots, u_{n})^{T}, 0 ⩽ τ_{i} (t) ⩽ τ, i = 1, 2, \dots, n . \end{matrix}$

The initial conditions for (2.1) are of the form ϕ = (ϕ₁, ϕ₂, … , ϕ_n), where each ϕ_i is a

Main results

In this section, some sufficient conditions of globally exponential stability for system (2.1) are obtained. Firstly, we give the following lemma due to [17]

Lemma 3.1

X^TY + Y^TX ⩽ X^TX + Y^TY. In particularly, if X and Y are vectors, $X^{T} Y ⩽ \frac{X^{T} X + Y^{T} Y}{2}$ .

About the existence of equilibrium point of system (2.1), we also have the following Lemma due to Lemma 1 in [16]

Lemma 3.2

Suppose that Assumption A, Assumption A, Assumption A are satisfied, then there exists an equilibrium point for neural system (2.1).

Secondly, we present two

Comparison and example

We now compare our results with the previous results derived in the literature, which are restated in the following theorems.

Theorem 4.1

[3]

Suppose that in system (3.1), τ_j(t) = τ_j (j = 1, 2, … , n), and Assumption A, Assumption A, Assumption A are satisfied. The origin of neural system (3.1) is globally asymptotically stable if there exists a symmetric positive diagonal matrices P = diag(p₁, p₂, … , p_n) and Q = diag(q₁, q₂, … , q_n) such that $Ω = 2 P Γ G^{- 1} - PA - A^{T} P - Q - {PBQ}^{- 1} B^{T} P > 0 .$

Theorem 4.2

[8]

Suppose that in system (3.1), τ_j(t) = τ_j (j = 1, 2, … , n), and

Conclusion

In this letter, we presented new results for the globally exponential stability properties of Cohen–Grossberg neural networks with time-varying delays. The global exponential stability criteria obtained in this paper have been derived by employing the Lyapunov functional method and the technique of inequality of integral. Comparisons between our results and the previous results have also been made. It was shown that our results establish a new set of global exponential stability criteria for a

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Short communicationGlobal exponential stability analysis of Cohen–Grossberg neural network with delays

Abstract

Introduction

Section snippets

Cohen–Grossberg neural network model and some basic concepts

Main results

Comparison and example

[3]

[8]

Conclusion

Neural Networks

Phys Lett A

J Math Anal Appl

Physica D: Nonlinear Phenom

Phys Lett A

Phys Lett A

Phys Lett A

Phys Lett A

Phys Lett A

Phys Lett A

Short communication
Global exponential stability analysis of Cohen–Grossberg neural network with delays