New LMI-based conditions for global exponential stability to a class of Cohen–Grossberg BAM networks with delays

doi:10.1016/j.neucom.2013.05.016

Neurocomputing

Volume 121, 9 December 2013, Pages 512-522

https://doi.org/10.1016/j.neucom.2013.05.016 Get rights and content

Abstract

This paper discusses global exponential stability of equilibrium point for a class of Cohen–Grossberg BAM neural networks with delays. Under the assumptions that the activation functions only satisfy global Lipschitz conditions and the behaved functions only satisfy sign conditions, by applying the linear matrix inequality (LMI) method, degree theory and some inequality technique, a novel LMI-based sufficient condition is established for global exponential stability of the concerned neural networks. In our result, the assumption on the activation functions is less conservative than the assumption for monotonicity in Nie and Cao (2009) [28] and the assumption on the behaved functions is also less conservative than the assumption for differentiability in Nie and Cao (2009) [28], Xia (2010) [30], Zhou and Wan (2009) [31] and Zhang et al. (2012) [35].

Section snippets

Preliminaries

In 1983, Cohen and Grossberg [1] constructed a kind of simplified neural networks which are now called as Cohen–Grossberg neural networks (CGNNS). These Cohen–Grossberg neural networks were designed to include Hopfield neural networks, shunting neural networks and some ecological systems. Cohen–Grossberg neural networks have received increasing interest due to their promising potential applications in many fields such as pattern recognition, parallel computing, associative memory, and

Preliminaries

Throughout this paper, we make the following notations: $| x | sign (x) = {(| x_{1} | sign (x_{1}), | x_{2} | sign (x_{2}), \dots, | x_{m} | sign (x_{m}))}^{T},$ $| y | sign (y) = {(| y_{1} | sign (y_{1}), | y_{2} | sign (y_{2}), \dots, | y_{m} | sign (y_{m}))}^{T},$ $| x (t - δ) - x^{⁎} | sign (x (t - δ) - x^{⁎}) = {(| x_{1} (t - δ) - x_{1}^{⁎} | sign (x_{1} (t - δ) - x_{1}^{⁎}), | x_{2} (t - δ) - x_{2}^{⁎} | \times sign (x_{2} (t - δ) - x_{2}^{⁎}), \dots, \times | x_{m} (t - δ) - x_{m}^{⁎} | sign (x_{m} (t - δ) - x_{m}^{⁎}))}^{T},$ $| y (t - τ) - y^{⁎} | sign (y (t - τ) - y^{⁎}) = {(| y_{1} (t - τ) - y_{1}^{⁎} | sign (y_{1} (t - τ) - y_{1}^{⁎}), | y_{2} (t - τ) - y_{2}^{⁎} | sign (y_{2} (t - τ) - y_{2}^{⁎}), \dots, | y_{m} (t - τ) - y_{m}^{⁎} | sign (y_{m} (t - τ) - y_{m}^{⁎}))}^{T},$ where $sign (x)$ is defined as when $x > 0, sign (x) = 1$ ; when $x < 0, sign (x) = - 1$ , when $x = 0, sign$

Global exponential stability

In this section, we will present our results on global exponential stability for system (1.3) by applying degree theory, LMI method, inequality technique and Lyapunov functional.

Theorem 3.1

Under assumptions $(H_{1}) - (H_{3})$ , system (1.3) has one unique equilibrium point which is globally exponentially stable if there exist m order positive definite diagonal matrices $P, M, P_{1} = (p_{1 i}), M_{1} = (m_{1 i}), Y_{i} (i = 1, 2, 3, 4)$ , and m order matrices $P_{2} = (p_{2 ij}), P_{3} = (p_{3 ij}), M_{2} = (m_{2 ij}), M_{3} = (m_{3 ij})$ and scalar $γ > 0$ such that the following linear

An example

Example 1

Consider the following Cohen–Grossberg BAM neural networks with delays: $\frac{d x_{i} (t)}{d t} = - a_{i} (x_{i} (t)) {b_{i} (x_{i} (t)) - \sum_{j = 1}^{m} s_{ij} f_{j} (y_{j} (t - τ)) + I_{i}}, i = 1, 2, \frac{d y_{j} (t)}{d t} = - c_{j} (y_{j} (t)) {d_{j} (y_{j} (t)) - \sum_{i = 1}^{m} t_{ji} g_{i} (x_{i} (t - δ)) + J_{j}}, j = 1, 2,$ where $γ = 1, δ = 3, τ = 2, I_{1} = 1, I_{2} = 3, J_{1} = 5, J_{2} = 7,$ $a_{1} (x) = 5 + \sin x, s_{11} = 0.1, s_{12} = 0.2, s_{21} = 0.2,$ $s_{22} = 0.3, t_{11} = 0.3, t_{12} = 0.1, t_{21} = 0.1,$ $t_{22} = 0.5, a_{2} (x) = 3 + 2 \cos x, c_{1} (x) = 7 - 2.3 \cos x,$ $c_{2} (x) = 3 - 1.2 \sin x, b_{i} (x) = d_{i} (x) = 6 x + | \cos x |,$ $f_{1} (y) = 0.1 (| y + 1 | + | y - 1 |), f_{2} (y) = 0.2 (| y + 1 | + | y - 1 |),$ $g_{1} (x) = 0.3 (| x + 1 | + | x - 1 |), g_{2} (x) = 0.4 (| x + 1 | + | x - 1 |) .$ Since for $\forall x, y \in R$ $sign (x - y) [b_{i} (x) - b_{i}$

Conclusion

In this paper, a novel LMI-based condition on global exponential stability for system (1.3) is obtained under the assumptions that the behaved functions only satisfy sign conditions and the activation functions only satisfy global Lipschitz conditions by using degree theory, LMI method and some inequality technique. Compared with known results, in our paper, the assumption on the activation functions is less conservative than the assumptions for boundedness and monotonicity in existing papers

Dongming Zhou He was born in 1963 and is a Professor of Information College of Yunnan University in China. His field of study is neural networks theory and applications. So far, he has been an author of more than 50 papers.

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The Cohen–Grossberg-type BAM neural networks (CG-BAMNNs) have their promising potential for the tasks of parallel computation, associative memory and have great ability to solve difficult optimization problems. Recently, there have been many results on the stability in Lyapunov sense and convergence of equilibrium point of CG-BAMNNs with delays [2,3,8,17,21–23,26,27,44,60,66,68,69,74]. Lyapunov stability is one of the important properties of dynamic systems.
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Equilibria and Stability Analysis of Cohen-Grossberg BAM Neural Networks on Time Scale
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Shenghua Yu was born in 1966 and is a Professor of Applied Mathematics of Hunan University in China. His field of study is economic mathematics, neural networks theory and applications. So far, he has been an author of more than 30 papers.

Zhengqiu Zhang was born in 1963 and is a Professor of Applied Mathematics of Hunan University in China. His field of study is neural networks theory and applications. So far, he has been an author of more than 50 papers.

^☆: Project supported by the fund of National Social Science (No: 12 BTJ014) and by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry of China (No: 20091341) and the Fund of National Natural Science of China (No: 61065008).

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New LMI-based conditions for global exponential stability to a class of Cohen–Grossberg BAM networks with delays☆

Abstract

Section snippets

Preliminaries

Preliminaries

Global exponential stability

An example

Conclusion

Phys. Lett. A

Neural Networks

Neural Networks

Physica D

Neurocomputing

Phys. Lett. A

Neural Networks

Phys. Lett. A

Neurocomputing

Neurocomputing

Neurocomputing

Neurocomputing

Neurocomputing

Neurocomputing

Neurocomputing

Nonlinear Anal.

Appl. Math. Comput.

Chaos Solitons Fractals

Phys. Lett. A

Math. Comput. Modell.