随机时滞模糊细胞神经网络的指数稳定性分析
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摘要
细胞神经网络(CNNs)自从上世纪80年代由Chua和Yang提出以来,它在诸如模式识别、信号处理、并行计算和组合优化等领域有很大的应用。然而,由于其自身有一些缺点,学者们将模糊逻辑理论引入到CNNs,形成模糊细胞神经网络(FCNNs),这发挥了模糊集理论和细胞神经网络的优势。FCNNs在机器智能、控制、决策等方面有重要应用。这些应用都是依赖于其动力学行为,尤其是模型的稳定性。
在实际应用中,时滞、脉冲等因素对FCNNs的稳定性有较大影响。目前,有关时滞和脉冲FCNNs的稳定性研究取得了一些结果。事实上,大脑中的神经元细胞信息传递是一个随机过程,必须考虑在噪音环境下的FCNNs的稳定性。近年来,考虑随机作用于FCNNs稳定性的结果不是很多,有些模型还未做过讨论,正因如此,本文主要的工作是考虑了几类随机作用下的FCNNs的指数稳定性,包括变时滞、分布时滞和脉冲的情况。利用随机过程和分析的知识,得到了一些判断这几类随机模型指数稳定性的准则,弥补和丰富了这方面研究,具有一定的理论和实际意义。本文的组织结构如下：
第一章,介绍了CNNs和FCNNs的背景知识,以及有关它们稳定性的研究现状。
第二章,讨论了一类具有随机变时滞模糊细胞神经网络的均方指数稳定性,借助随机分析和构造Lyapunov泛函,得到了确保这类系统平衡点均方指数稳定性的一个充分条件,所得的充分条件易于满足,数值例子证实了结果的有效性。
第三章,利用Lyapunov泛函,Ito's公式,不等式技巧和非负半鞅收敛定理研究了一类随机分布时滞模糊细胞神经网络,获得了一个使得这类系统平衡点几乎必然指数稳定和均方指数稳定的一个充分条件。举例说明了结论的有效性。
第四章,借助Lyapunov泛函,Ito's公式,不等式等分析方法讨论了一类随机脉冲分布时滞模糊细胞神经网络的均方指数稳定性,得到了一个使这类系统平衡点均方指数稳定的充分判据,弥补了这类随机模型稳定性研究的空白。同时给了一些推论和注记,大大推广了前人的结果,最后通过一个例子说明了结论的正确性。
Since cellar neural networks(CNNs) was introduced in 1980s by Chua and Yang, this model has received increasing interest due to its promising potential applications in many fields such as pattern recognition, signal processing, parallel computing and combinatorial optimization. However, there have shortcomings in itself. Some researchers introduced fuzzy logical theory into cellar neural networks to form fuzzy cellar neural networks(FCNNs), which as results, maximizes the advantages of cellar neural networks and fuzzy set theory. FCNNs have potential application in many fields such as machine intelligence, control, decision analysis and so on. These applications heavily depend on the dynamical behaviors, especially the stability of neural networks.
In practical application, some factors that have great influence on the stability of FCNNs such as delay and impulsion. At present, some results about the stabil-ity of FCNNs with delays and impulsion are obtained. In fact, the transmission of neurons cell in the brains is noisy process. So, we should consider the stability of FCNNs in the noise environments. In recent years, there are few about stochastic effect on the stability of FCNNs. Furthermore, several type of stochastic FCNNs haven't been discussed by now. The paper mainly consider the stability of FCNNs with stochastic perturbation, which including time—varying delays, distributed de-lays and impulsion. By using stochastic processes and analysis, some criteria to judge the the stability of FCNNs affected by stochastic perturbation are obtained, which make up and enrich the result in this areas and has some theoretical and practical value. The paper is organized as follows:
In chapter 1, we mainly focus on the background of CNNS and FCNNs, and research status of their stability.
In chapter 2, a class of stochastic fuzzy cellular neural networks with time—varying delays is considered. Sufficient conditions for the mean square exponential stability are obtained by using Lyapunov functional and stochastic analysis, which is easy to satisfy. An example is provided to demonstrate the usefulness of the proposed criteria.
In chapter 3, the exponential stability of a class of stochastic fuzzy cellular neural networks with distributed delays is investigated in this paper. By using an-alytic methods such as Lyapunov functional, Ito's formula, inequality techniques and nonegative semimartingale convergence theorem, the sufficient conditions guar-anteeing the almost sure and mean square exponential stability of its equilibrium solution are respectively obtained. For illustration, an example is given to show the feasibility of results.
In chapter 4, by using analytic methods such as Lyapunov functional, Ito's formula, inequality techniques. The mean square exponential stability of a class of impulsive stochastic fuzzy cellular neural networks with distributed delays is inves-tigated, the sufficient conditions guaranteeing the mean square exponential stability of its equilibrium solution are obtained, which make up for gaps in stability of such stochastic models, At the same time, some inferences and remarks were prospered which greatly extend the previous results. At last the conclusion is illuminated through an example.
在实际应用中,时滞、脉冲等因素对FCNNs的稳定性有较大影响。目前,有关时滞和脉冲FCNNs的稳定性研究取得了一些结果。事实上,大脑中的神经元细胞信息传递是一个随机过程,必须考虑在噪音环境下的FCNNs的稳定性。近年来,考虑随机作用于FCNNs稳定性的结果不是很多,有些模型还未做过讨论,正因如此,本文主要的工作是考虑了几类随机作用下的FCNNs的指数稳定性,包括变时滞、分布时滞和脉冲的情况。利用随机过程和分析的知识,得到了一些判断这几类随机模型指数稳定性的准则,弥补和丰富了这方面研究,具有一定的理论和实际意义。本文的组织结构如下：
第一章,介绍了CNNs和FCNNs的背景知识,以及有关它们稳定性的研究现状。
第二章,讨论了一类具有随机变时滞模糊细胞神经网络的均方指数稳定性,借助随机分析和构造Lyapunov泛函,得到了确保这类系统平衡点均方指数稳定性的一个充分条件,所得的充分条件易于满足,数值例子证实了结果的有效性。
第三章,利用Lyapunov泛函,Ito's公式,不等式技巧和非负半鞅收敛定理研究了一类随机分布时滞模糊细胞神经网络,获得了一个使得这类系统平衡点几乎必然指数稳定和均方指数稳定的一个充分条件。举例说明了结论的有效性。
第四章,借助Lyapunov泛函,Ito's公式,不等式等分析方法讨论了一类随机脉冲分布时滞模糊细胞神经网络的均方指数稳定性,得到了一个使这类系统平衡点均方指数稳定的充分判据,弥补了这类随机模型稳定性研究的空白。同时给了一些推论和注记,大大推广了前人的结果,最后通过一个例子说明了结论的正确性。
Since cellar neural networks(CNNs) was introduced in 1980s by Chua and Yang, this model has received increasing interest due to its promising potential applications in many fields such as pattern recognition, signal processing, parallel computing and combinatorial optimization. However, there have shortcomings in itself. Some researchers introduced fuzzy logical theory into cellar neural networks to form fuzzy cellar neural networks(FCNNs), which as results, maximizes the advantages of cellar neural networks and fuzzy set theory. FCNNs have potential application in many fields such as machine intelligence, control, decision analysis and so on. These applications heavily depend on the dynamical behaviors, especially the stability of neural networks.
In practical application, some factors that have great influence on the stability of FCNNs such as delay and impulsion. At present, some results about the stabil-ity of FCNNs with delays and impulsion are obtained. In fact, the transmission of neurons cell in the brains is noisy process. So, we should consider the stability of FCNNs in the noise environments. In recent years, there are few about stochastic effect on the stability of FCNNs. Furthermore, several type of stochastic FCNNs haven't been discussed by now. The paper mainly consider the stability of FCNNs with stochastic perturbation, which including time—varying delays, distributed de-lays and impulsion. By using stochastic processes and analysis, some criteria to judge the the stability of FCNNs affected by stochastic perturbation are obtained, which make up and enrich the result in this areas and has some theoretical and practical value. The paper is organized as follows:
In chapter 1, we mainly focus on the background of CNNS and FCNNs, and research status of their stability.
In chapter 2, a class of stochastic fuzzy cellular neural networks with time—varying delays is considered. Sufficient conditions for the mean square exponential stability are obtained by using Lyapunov functional and stochastic analysis, which is easy to satisfy. An example is provided to demonstrate the usefulness of the proposed criteria.
In chapter 3, the exponential stability of a class of stochastic fuzzy cellular neural networks with distributed delays is investigated in this paper. By using an-alytic methods such as Lyapunov functional, Ito's formula, inequality techniques and nonegative semimartingale convergence theorem, the sufficient conditions guar-anteeing the almost sure and mean square exponential stability of its equilibrium solution are respectively obtained. For illustration, an example is given to show the feasibility of results.
In chapter 4, by using analytic methods such as Lyapunov functional, Ito's formula, inequality techniques. The mean square exponential stability of a class of impulsive stochastic fuzzy cellular neural networks with distributed delays is inves-tigated, the sufficient conditions guaranteeing the mean square exponential stability of its equilibrium solution are obtained, which make up for gaps in stability of such stochastic models, At the same time, some inferences and remarks were prospered which greatly extend the previous results. At last the conclusion is illuminated through an example.
引文
[1]Chua, Leon O., Yang, Lin, Cellular neural networks:Theory, IEEE transactions on circuits and systems 35(10),1257-1272,1988.
[2]Chua, Leon 0., Yang, Lin, Cellular neural networks:Applications, IEEE transactions on circuits and systems 35(10),1273-1290,1988.
[3]Chua, Leon O., Roska, Tamas, Stability of a class of nonreciprocal cellular neural net-works, IEEE transactions on circuits and systems 37(12),1520-1527,1990.
[4]Civalleri, Pier Paolo, Gilli, Marco, Pandolfi., Luciano, On stability of cellular neural networks with delay, IEEE Transactions on Circuits and Systems I:Fundamental Theory and Applications 40(3),157-165,1993.
[5]Guzelis, Cuneyt, Chua, Leon O., Stability analysis of generalized cellular neural net-works, International Journal of Circuit Theory and Applications 21(1),1-33,1993.
[6]Liao, T., Wang, F., Global stability condition for cellular neural networks with delay, Electronics Letters 35(16),1347-1349,1999.
[7]Shao, H., Less conservative delay-dependent stability criteria for neural networks with time-varying delays Neurocomputing 73(7-9),1528-1532,2010.
[8]Zhao, W., Zhu, Q, New results of global robust exponential stability of neural networks with delays Nonlinear Analysis:Real World Applications 11(2),1190-1197,2010.
[9]Cao, J., On stability of delayed cellular neural networks, Physics Letters, Section A: General, Atomic and Solid State Physics 261(5-6),303-308,1999.
[10]Ma, K.,Yu, Li.,Zhang, W. Global exponential stability of cellular neural networks with time-varying discrete and distributed delays, Neurocomputing 72(10-12),2705-2709,2009.
[11]Singh, V., Robust stability of cellular neural networks with delay:Linear matrix in-equality approach, IEE Proceedings:Control Theory and Applications 151(1),125-129, 2004.
[12]Lou, X., Cui, B., Robust exponential stabilization of a class of delayed neural networks with reaction-diffusion terms,International Journal of Neural Systems 16(6),435-443, 2006.
[13]Zhao, H., Mao, Z.,Boundedness and stability of nonautonomous cellular neural net-works with reaction-diffusion terms, Mathematics and Computers in Simulation 79(5), 1603-1617,2009.
[14]Li, K., Zhang, X., Li, Z., Global exponential stability of impulsive cellular neural networks with time-varying and distributed delay, Chaos, Solitons & Fractals,41(3), 1427-1434,2009.
[15]Li, D., Yang, D., Wang, H., Zhang, X., Wang, S., Asymptotical stability of multi-delayed cellular neural networks with impulsive effects, Physica A:Statistical Mechanics and its Applications 388(2-3),218-224,2009.
[16]Huang, Z., Yang, Q., Exponential stability of impulsive high-order cellular neural net-works with time-varying delays Nonlinear Analysis:Real World Applications 11(1), 592-600,2010.
[17]Huang, Z., Yang, Q., Existence and exponential stability of almost periodic solution for stochastic cellular neural networks with delay, Chaos, Solitons & Fractals 42(2),773-780, 2009.
[18]Zhang, X., Wang, G., Liu, H., Stability analysis of delayed cellular neural networks with and without noise perturbation,Applied Mathematics and Mechanics (English Edition) 29(11),1427-1438,2008.
[19]Lu, J., Ma, Y., Mean square exponential stability and periodic solutions of stochastic delay cellular neural networks, Chaos, Solitons & Fractals,38(5),1323-1331,2008.
[20]Yang, T, Yang, L., Wu, Chai Wah, Chua, Leon 0., Fuzzy cellular neural networks: applications, Proceedings of the IEEE International Workshop on Cellular Neural Net-works and their Applications,225-230,1996.
[21]Yang, T, Yang, L., Wu, Chai Wah, Chua, Leon 0., Fuzzy cellular neural networks: theory, Proceedings of the IEEE International Workshop on Cellular Neural Networks and their Applications,181-186,1996.
[22]Liu, Y., Tang, W., Exponential stability of fuzzy cellular neural networks with constant and time-varying delays,Physics Letters, Section A:General, Atomic and Solid State Physics 323(3-4),224-233,2004.
[23]Chen, Y., Liao, X., Novel exponential stability criteria for fuzzy cellular neural networks with time-varying delay, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) 3173,120-125,2004.
[24]Huang, T., Exponential stability of fuzzy cellular neural networks with distributed delay, Physics Letters, Section A:General, Atomic and Solid State Physics 351(1-2), 48-52,2006.
[25]Zhang, Q., Xiang, R., Global asymptotic stability of fuzzy cellular neural networks with time-varying delays, Physics Letters, Section A:General, Atomic and Solid State Physics 372(22),3971-3977,2008.
[26]Yuan, K., Cao, J., Deng, J., Exponential stability and periodic solutions of fuzzy cellular neural networks with time-varying delays, Neurocomputing 69(13-15),1619-1627,2006.
[27]Huang, T., Roque-Sol, M., Stability of fuzzy cellular neural networks with impulses, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial In-telligence and Lecture Notes in Bioinformatics) 3971 LNCS,243-248,2006.
[28]Rakkiyappan, R, Balasubramaniam, P, On exponential stability results for fuzzy im-pulsive neural networks,Fuzzy Sets and Systems 161(13),2010,1823-1835.
[29]Wang, J., Lu, J.G., Global exponential stability of fuzzy cellular neural networks with delays and reaction-diffusion terms, Chaos, Solitons and Fractals 38(3),878-885,2008.
[30]Chen, L., Zhao, H., Stability analysis of stochastic fuzzy cellular neural networks with delays, Neurocomputing 72(1-3),436-444,2008.
[31]Zhao, H., Ding, N., Chen, L., Almost sure exponential stability of stochastic fuzzy cellular neural networks with delays, Chaos, Solitons & Fractals 40(4),1653-1659,2009.
[32]Wang, X., Xu, D., Global exponential stability of impulsive fuzzy cellular neural net-works with mixed delays and reaction-diffusion terms, Chaos, Solitons and Fractals 42(5),2713-2721,2009.
[33]Li, Z., Li, K., Stability analysis of impulsive fuzzy cellular neural networks with dis-tributed delays and reaction-diffusion terms, Chaos, Solitons & Fractals,42(1),492-499, 2009.
[34]Yang, H., Sheng, L., Robust stability of uncertain stochastic fuzzy cellular neural networks, Neurocomputing 73(1-3),133-138,2009.
[35]Cao,J., Wang, J.,Absolute exponential stability of recurrent neural networks with Lipschitz-continuous activation functions and time delays. Neural networks,17(3),379-390,2004.
[36]Mao,X. Exponential Stability of Stochastic Dif-ferential Equations,Marcel Dekker, NY. 1994.
[37]Mao, X. Stochastic Dierential Equationsand Applications, Horwood Publishing,1997.
[38]Wang, X., Guo, Q., Xu, D, Exponential p-stability of impulsive stochastic Cohen-Grossberg neural networks with mixed delays, Mathematics and Computers in Simulation 79(5),1698-1710,2009.
[39]Xu, L., Xu, D., Exponential p-stability of impulsive stochastic neural networks with mixed delays, Chaos, Solitons & Fractals,41(1),263-272,2009.
[40]Song, Q., Wang, Z., Stability analysis of impulsive stochastic Cohen-Grossberg neural networks with mixed time delays, Physica A 387(13),3314-3326,2008.
[41]Song, Q., Cao, J., Dynamical behaviors of discrete-time fuzzy cellular neural networks with variable delays and impulsives, J. Franklin Inst 345(1),39-59,2008.
[42]Li, Z., Li, K., Stability analysis of impulsive Cohen-Grossberg neural networks with distributed delays and reaction-diffusion terms, Applied Mathematical Modelling 33(3), 1337-1348,2009.
[2]Chua, Leon 0., Yang, Lin, Cellular neural networks:Applications, IEEE transactions on circuits and systems 35(10),1273-1290,1988.
[3]Chua, Leon O., Roska, Tamas, Stability of a class of nonreciprocal cellular neural net-works, IEEE transactions on circuits and systems 37(12),1520-1527,1990.
[4]Civalleri, Pier Paolo, Gilli, Marco, Pandolfi., Luciano, On stability of cellular neural networks with delay, IEEE Transactions on Circuits and Systems I:Fundamental Theory and Applications 40(3),157-165,1993.
[5]Guzelis, Cuneyt, Chua, Leon O., Stability analysis of generalized cellular neural net-works, International Journal of Circuit Theory and Applications 21(1),1-33,1993.
[6]Liao, T., Wang, F., Global stability condition for cellular neural networks with delay, Electronics Letters 35(16),1347-1349,1999.
[7]Shao, H., Less conservative delay-dependent stability criteria for neural networks with time-varying delays Neurocomputing 73(7-9),1528-1532,2010.
[8]Zhao, W., Zhu, Q, New results of global robust exponential stability of neural networks with delays Nonlinear Analysis:Real World Applications 11(2),1190-1197,2010.
[9]Cao, J., On stability of delayed cellular neural networks, Physics Letters, Section A: General, Atomic and Solid State Physics 261(5-6),303-308,1999.
[10]Ma, K.,Yu, Li.,Zhang, W. Global exponential stability of cellular neural networks with time-varying discrete and distributed delays, Neurocomputing 72(10-12),2705-2709,2009.
[11]Singh, V., Robust stability of cellular neural networks with delay:Linear matrix in-equality approach, IEE Proceedings:Control Theory and Applications 151(1),125-129, 2004.
[12]Lou, X., Cui, B., Robust exponential stabilization of a class of delayed neural networks with reaction-diffusion terms,International Journal of Neural Systems 16(6),435-443, 2006.
[13]Zhao, H., Mao, Z.,Boundedness and stability of nonautonomous cellular neural net-works with reaction-diffusion terms, Mathematics and Computers in Simulation 79(5), 1603-1617,2009.
[14]Li, K., Zhang, X., Li, Z., Global exponential stability of impulsive cellular neural networks with time-varying and distributed delay, Chaos, Solitons & Fractals,41(3), 1427-1434,2009.
[15]Li, D., Yang, D., Wang, H., Zhang, X., Wang, S., Asymptotical stability of multi-delayed cellular neural networks with impulsive effects, Physica A:Statistical Mechanics and its Applications 388(2-3),218-224,2009.
[16]Huang, Z., Yang, Q., Exponential stability of impulsive high-order cellular neural net-works with time-varying delays Nonlinear Analysis:Real World Applications 11(1), 592-600,2010.
[17]Huang, Z., Yang, Q., Existence and exponential stability of almost periodic solution for stochastic cellular neural networks with delay, Chaos, Solitons & Fractals 42(2),773-780, 2009.
[18]Zhang, X., Wang, G., Liu, H., Stability analysis of delayed cellular neural networks with and without noise perturbation,Applied Mathematics and Mechanics (English Edition) 29(11),1427-1438,2008.
[19]Lu, J., Ma, Y., Mean square exponential stability and periodic solutions of stochastic delay cellular neural networks, Chaos, Solitons & Fractals,38(5),1323-1331,2008.
[20]Yang, T, Yang, L., Wu, Chai Wah, Chua, Leon 0., Fuzzy cellular neural networks: applications, Proceedings of the IEEE International Workshop on Cellular Neural Net-works and their Applications,225-230,1996.
[21]Yang, T, Yang, L., Wu, Chai Wah, Chua, Leon 0., Fuzzy cellular neural networks: theory, Proceedings of the IEEE International Workshop on Cellular Neural Networks and their Applications,181-186,1996.
[22]Liu, Y., Tang, W., Exponential stability of fuzzy cellular neural networks with constant and time-varying delays,Physics Letters, Section A:General, Atomic and Solid State Physics 323(3-4),224-233,2004.
[23]Chen, Y., Liao, X., Novel exponential stability criteria for fuzzy cellular neural networks with time-varying delay, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) 3173,120-125,2004.
[24]Huang, T., Exponential stability of fuzzy cellular neural networks with distributed delay, Physics Letters, Section A:General, Atomic and Solid State Physics 351(1-2), 48-52,2006.
[25]Zhang, Q., Xiang, R., Global asymptotic stability of fuzzy cellular neural networks with time-varying delays, Physics Letters, Section A:General, Atomic and Solid State Physics 372(22),3971-3977,2008.
[26]Yuan, K., Cao, J., Deng, J., Exponential stability and periodic solutions of fuzzy cellular neural networks with time-varying delays, Neurocomputing 69(13-15),1619-1627,2006.
[27]Huang, T., Roque-Sol, M., Stability of fuzzy cellular neural networks with impulses, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial In-telligence and Lecture Notes in Bioinformatics) 3971 LNCS,243-248,2006.
[28]Rakkiyappan, R, Balasubramaniam, P, On exponential stability results for fuzzy im-pulsive neural networks,Fuzzy Sets and Systems 161(13),2010,1823-1835.
[29]Wang, J., Lu, J.G., Global exponential stability of fuzzy cellular neural networks with delays and reaction-diffusion terms, Chaos, Solitons and Fractals 38(3),878-885,2008.
[30]Chen, L., Zhao, H., Stability analysis of stochastic fuzzy cellular neural networks with delays, Neurocomputing 72(1-3),436-444,2008.
[31]Zhao, H., Ding, N., Chen, L., Almost sure exponential stability of stochastic fuzzy cellular neural networks with delays, Chaos, Solitons & Fractals 40(4),1653-1659,2009.
[32]Wang, X., Xu, D., Global exponential stability of impulsive fuzzy cellular neural net-works with mixed delays and reaction-diffusion terms, Chaos, Solitons and Fractals 42(5),2713-2721,2009.
[33]Li, Z., Li, K., Stability analysis of impulsive fuzzy cellular neural networks with dis-tributed delays and reaction-diffusion terms, Chaos, Solitons & Fractals,42(1),492-499, 2009.
[34]Yang, H., Sheng, L., Robust stability of uncertain stochastic fuzzy cellular neural networks, Neurocomputing 73(1-3),133-138,2009.
[35]Cao,J., Wang, J.,Absolute exponential stability of recurrent neural networks with Lipschitz-continuous activation functions and time delays. Neural networks,17(3),379-390,2004.
[36]Mao,X. Exponential Stability of Stochastic Dif-ferential Equations,Marcel Dekker, NY. 1994.
[37]Mao, X. Stochastic Dierential Equationsand Applications, Horwood Publishing,1997.
[38]Wang, X., Guo, Q., Xu, D, Exponential p-stability of impulsive stochastic Cohen-Grossberg neural networks with mixed delays, Mathematics and Computers in Simulation 79(5),1698-1710,2009.
[39]Xu, L., Xu, D., Exponential p-stability of impulsive stochastic neural networks with mixed delays, Chaos, Solitons & Fractals,41(1),263-272,2009.
[40]Song, Q., Wang, Z., Stability analysis of impulsive stochastic Cohen-Grossberg neural networks with mixed time delays, Physica A 387(13),3314-3326,2008.
[41]Song, Q., Cao, J., Dynamical behaviors of discrete-time fuzzy cellular neural networks with variable delays and impulsives, J. Franklin Inst 345(1),39-59,2008.
[42]Li, Z., Li, K., Stability analysis of impulsive Cohen-Grossberg neural networks with distributed delays and reaction-diffusion terms, Applied Mathematical Modelling 33(3), 1337-1348,2009.