汪菁宜++彭國強

摘 要 本文旨在研究一類帶變時滯的隨機模糊細胞神經(jīng)網(wǎng)絡的穩(wěn)定性.通過構(gòu)造恰當?shù)腖yapunov泛函并運用線性矩陣不等式（LMI）理論，作者給出了保證這類神經(jīng)網(wǎng)絡全局漸近穩(wěn)定的充分條件.本文推導出兩個定理：一個用以判定文中模型的全局漸進穩(wěn)定性，一個用以判定該模型在均方意義下的全局漸近穩(wěn)定性.

關(guān)鍵詞 隨機模糊神經(jīng)網(wǎng)絡；變時滯；全局漸近穩(wěn)定性

Globally Asymptotic Stability of Stochastic Fuzzy Cellular

Neural Networks with Timevarying Delays

WANG Jingyi， PENG Guoqiang

（College of Mathematics & Econometrics， Hunan University， Changsha， Hunan 410082，China）

Abstract This paper aims at solving the problem of checking the stability of a class of stochastic fuzzy cellular neural networks with timevarying delays. By constructing suitable Lyapunov functional and applying linear matrix inequality（LMI） theory， some sufficient conditions were developed to guarantee its globally asymptotic stability of this kind of neural networks. Two main results were obtained： one considering the globally asymptotic stability of the model， the other regarding its globally asymptotic stability in the mean square.

Key words stochastic fuzzy neural networks； timevarying delays； globally asymptotic stability

中圖分類號 O29 文獻標識碼 A

1 Introduction

It is well known that fuzzy cellular neural networks（FCNN） which integrates fuzzy logic into traditional cellular neural networks brought up by Chua and Yang in 1988 have become a useful tool in a lot of fields like signal processing， pattern recognition， associative memory and image processing[1-3]. Furthermore， time delays are frequently encountered in hardware implementation and they can destroy a stable network and cause oscillations， bifurcation and chaos. Thus， it is of great importance to study the stability of delayed fuzzy cellular neural networks. In actuality， a great deal of studies focusing on this issue have emerged in recent years[4-6].

However， a real system is usually affected by external perturbations and hence should be treated as random. In fact， the synaptic transmission is a noisy process caused by random fluctuations from the release of neurotransmitters and other probabilistic factors. Moreover， a neural network can be stabilized or destabilized by certain stochastic inputs[7]. Accordingly， the study for stability of stochastic FCNNs becomes urgent and consequently some results have been derived[8-11].

The main purpose of this paper is to study the globally asymptotic stability of a kind of stochastic fuzzy cellular neural networks with timevarying delays. We tried to derive our results by applying the linear matrix inequality（LMI） approach， which， to the authors best knowledge， has not been used on this kind of systems before. Our conditions for stability are expressed in terms of linear matrix inequalities which can be easily solved by some standard numerical packages.endprint

Thus， the proof is completed.

Theorem 2 Under the same conditions of Theorem 1， system （1） is globally asymptotic stable in the mean square.

By the stability results in [7]， the neural network （1） is globally asymptotically stable in the mean square.

5 Conclusion

In this paper， the sufficient conditions have been derived for checking the globally asymptotic stability and the globally asymptotic stability in the mean square of a class of stochastic fuzzy cellular neural networks with timevarying delays by constructing suitable Lyapunov functional and applying LMI approach. Besides， a numerical example has been given to testify the effectiveness of our methods.

References

[1] S HAYKIN. Neural Networks： A Comprehensive Foundation[M].Commun. in Partial Diff Eqns， 1994：105-143.

[2] L O CHUA， L YANG. Cellular neural networks： applications[J].IEEE Trans. Circuits Systems， 1988，35（10）：1257-1272.

[3] T YANG， Y YANG， C WU， et al. Fuzzy cellular neural networks： theory[J].Proceedings of IEEE international workshop on mathematical morphological operations，1996：225-230.

[4] Y LIU， W TANG. Exponential stability of fuzzy cellular neural networks with constant and timevarying delays[J].Phys Letts A， 2004， 323：224-233.

[5] Q ZHANG， R XIANG. Global asymptotic stability of fuzzy cellular neural networks with timevaring delays[J].Phys. Letts. A， 2008， 372：3971-3977.

[6] K YUAN， J CAO， J DENG. Exponential stability and periodic solutions of fuzzy cellular neural networks with timevaring delays[J].Neurocomputing， 2006， 69：1619-1627.

[7] X MAO. Stochastic differential equations and applications[M].Chichester： Horwood Publishing，2007：110-127.

[8] S BLYTHE， X MAO，X LIAO. Stability of stochastic delay neural networks[M].J Franklin Inst， 2001：338-481.

[9] H ZHAO， N DING， L CHEN. Almost sure exponential stability of stochastic fuzzy cellular neural networks with delays[J].Chaos， Solitons and Fractals， 2009：1653-1659.

[10]L CHEN， R WU， D PAN. Mean square exponential stability of impulsive stochastic fuzzy cellular neural networks with distributed delays[J].Expert Systems with Applications， 2011，38：6294-6299.

[11]M Syed ALI， P BALASUBRAMANIAM. Global asymptotic stability of stochastic fuzzy cellular neural networks with multiple discrete and distributed time-varing delays.[J]Commun Nonlinear Sci Numer Simulat， 2010：1155-1167.endprint

Thus， the proof is completed.

Theorem 2 Under the same conditions of Theorem 1， system （1） is globally asymptotic stable in the mean square.

By the stability results in [7]， the neural network （1） is globally asymptotically stable in the mean square.

5 Conclusion

In this paper， the sufficient conditions have been derived for checking the globally asymptotic stability and the globally asymptotic stability in the mean square of a class of stochastic fuzzy cellular neural networks with timevarying delays by constructing suitable Lyapunov functional and applying LMI approach. Besides， a numerical example has been given to testify the effectiveness of our methods.

References

[1] S HAYKIN. Neural Networks： A Comprehensive Foundation[M].Commun. in Partial Diff Eqns， 1994：105-143.

[2] L O CHUA， L YANG. Cellular neural networks： applications[J].IEEE Trans. Circuits Systems， 1988，35（10）：1257-1272.

[3] T YANG， Y YANG， C WU， et al. Fuzzy cellular neural networks： theory[J].Proceedings of IEEE international workshop on mathematical morphological operations，1996：225-230.

[4] Y LIU， W TANG. Exponential stability of fuzzy cellular neural networks with constant and timevarying delays[J].Phys Letts A， 2004， 323：224-233.

[5] Q ZHANG， R XIANG. Global asymptotic stability of fuzzy cellular neural networks with timevaring delays[J].Phys. Letts. A， 2008， 372：3971-3977.

[6] K YUAN， J CAO， J DENG. Exponential stability and periodic solutions of fuzzy cellular neural networks with timevaring delays[J].Neurocomputing， 2006， 69：1619-1627.

[7] X MAO. Stochastic differential equations and applications[M].Chichester： Horwood Publishing，2007：110-127.

[8] S BLYTHE， X MAO，X LIAO. Stability of stochastic delay neural networks[M].J Franklin Inst， 2001：338-481.

[9] H ZHAO， N DING， L CHEN. Almost sure exponential stability of stochastic fuzzy cellular neural networks with delays[J].Chaos， Solitons and Fractals， 2009：1653-1659.

[10]L CHEN， R WU， D PAN. Mean square exponential stability of impulsive stochastic fuzzy cellular neural networks with distributed delays[J].Expert Systems with Applications， 2011，38：6294-6299.

[11]M Syed ALI， P BALASUBRAMANIAM. Global asymptotic stability of stochastic fuzzy cellular neural networks with multiple discrete and distributed time-varing delays.[J]Commun Nonlinear Sci Numer Simulat， 2010：1155-1167.endprint

Thus， the proof is completed.

Theorem 2 Under the same conditions of Theorem 1， system （1） is globally asymptotic stable in the mean square.

By the stability results in [7]， the neural network （1） is globally asymptotically stable in the mean square.

5 Conclusion

In this paper， the sufficient conditions have been derived for checking the globally asymptotic stability and the globally asymptotic stability in the mean square of a class of stochastic fuzzy cellular neural networks with timevarying delays by constructing suitable Lyapunov functional and applying LMI approach. Besides， a numerical example has been given to testify the effectiveness of our methods.

References

[1] S HAYKIN. Neural Networks： A Comprehensive Foundation[M].Commun. in Partial Diff Eqns， 1994：105-143.

[2] L O CHUA， L YANG. Cellular neural networks： applications[J].IEEE Trans. Circuits Systems， 1988，35（10）：1257-1272.

[3] T YANG， Y YANG， C WU， et al. Fuzzy cellular neural networks： theory[J].Proceedings of IEEE international workshop on mathematical morphological operations，1996：225-230.

[4] Y LIU， W TANG. Exponential stability of fuzzy cellular neural networks with constant and timevarying delays[J].Phys Letts A， 2004， 323：224-233.

[5] Q ZHANG， R XIANG. Global asymptotic stability of fuzzy cellular neural networks with timevaring delays[J].Phys. Letts. A， 2008， 372：3971-3977.

[6] K YUAN， J CAO， J DENG. Exponential stability and periodic solutions of fuzzy cellular neural networks with timevaring delays[J].Neurocomputing， 2006， 69：1619-1627.

[7] X MAO. Stochastic differential equations and applications[M].Chichester： Horwood Publishing，2007：110-127.

[8] S BLYTHE， X MAO，X LIAO. Stability of stochastic delay neural networks[M].J Franklin Inst， 2001：338-481.

[9] H ZHAO， N DING， L CHEN. Almost sure exponential stability of stochastic fuzzy cellular neural networks with delays[J].Chaos， Solitons and Fractals， 2009：1653-1659.

[10]L CHEN， R WU， D PAN. Mean square exponential stability of impulsive stochastic fuzzy cellular neural networks with distributed delays[J].Expert Systems with Applications， 2011，38：6294-6299.

[11]M Syed ALI， P BALASUBRAMANIAM. Global asymptotic stability of stochastic fuzzy cellular neural networks with multiple discrete and distributed time-varing delays.[J]Commun Nonlinear Sci Numer Simulat， 2010：1155-1167.endprint