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      具時(shí)滯廣義CohenGrossberg神經(jīng)網(wǎng)絡(luò)的全局漸近穩(wěn)定性

      2014-06-24 18:49:20全志勇吳奇鋒
      經(jīng)濟(jì)數(shù)學(xué) 2014年1期
      關(guān)鍵詞:時(shí)滯全局單調(diào)

      全志勇++吳奇鋒

      摘 要 利用同胚映射原理、線性矩陣不等式和構(gòu)造的Lyapunov泛函研究了一類CohenGrossberg神經(jīng)網(wǎng)絡(luò)平衡點(diǎn)的全局漸近穩(wěn)定性,優(yōu)化了現(xiàn)有文獻(xiàn)中關(guān)于全局漸近穩(wěn)定性的判據(jù).

      關(guān)鍵詞 廣義CohenGrossberg神經(jīng)網(wǎng)絡(luò);全局漸近穩(wěn)定性;線性矩陣不等式;同胚

      中圖分類號(hào) O175.1 文獻(xiàn)標(biāo)識(shí)碼 A

      Global Asymptotic Stability of Generalized

      CohenGrossberg Neural Networks with Delays

      QUAN Zhiyong, WU Qifeng

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

      Abstract By means of Homeomorphism theory, linear matrix inequality and constructing a Lyapunov functional, we studied the global asymptotic stability of the equilibrium point for a class of CohenGrossberg neural networks. In our results, the criteria for the global asymptotical stability are better than that in existing papers.

      Key words generalized CohenGrossberg neural networks; global asymptotic stability; linear matrix inequality; Homeomorphism

      1 引 言

      由于CohenGrossberg神經(jīng)網(wǎng)絡(luò)(CGNN)在并行計(jì)算、聯(lián)想記憶,特別是最優(yōu)化計(jì)算等領(lǐng)域的重要作用,近年來,有或無時(shí)滯的 CGNN特別是一維CGNN的穩(wěn)定性問題已為國(guó)內(nèi)外學(xué)者所廣泛關(guān)注和研究,各種有趣的結(jié)果也被發(fā)表[1-6].然而,只有幾個(gè)作者討論了二維CGNN模型的穩(wěn)定性問題[7-10].在許多應(yīng)用中,由于二維CGNN考慮兩個(gè)神經(jīng)網(wǎng)絡(luò)之間的相互作用,因此對(duì)二維CGNN穩(wěn)定性的研究比對(duì)一維CGNN穩(wěn)定性的研究更有趣.這促使我們研究二維CGNN的穩(wěn)定性.

      本文將用不同于文獻(xiàn)[7]中的方法,即通過應(yīng)用同胚映射原理、不等式、線性矩陣不等式和構(gòu)造的Lyapunov泛函,對(duì)文獻(xiàn)[7]中具有多時(shí)滯的廣義二維CGNN的全局穩(wěn)定性繼續(xù)討論,得到了全局漸近穩(wěn)定性的新結(jié)果.當(dāng)把網(wǎng)絡(luò)降低為一維CGNN時(shí),獲得的的結(jié)果不同于現(xiàn)有文獻(xiàn)中的結(jié)果.在本文的結(jié)果中,去除了對(duì)行為函數(shù)在文獻(xiàn)[1-3]中的單調(diào)性假設(shè)和文獻(xiàn)[4,5]中的可微性假設(shè),對(duì)激勵(lì)函數(shù)去除了在文獻(xiàn)[1-5]中的有界性假設(shè)和文獻(xiàn)[2-5]中的單調(diào)性假設(shè).同討論的二維CGNN相比,在所得結(jié)果中,也去除了文獻(xiàn)[10]中對(duì)行為函數(shù)的單調(diào)性和可微性假設(shè)及對(duì)激勵(lì)函數(shù)的單調(diào)性假設(shè)和逆Lipschitz條件.由于用于研究全局漸近穩(wěn)定性的方法不同于文獻(xiàn)[7,8]中所用方法,因此關(guān)于全局漸近穩(wěn)定性的結(jié)果也不同于文獻(xiàn)[7,8]中所得到的結(jié)果.也就是說,在本文的結(jié)果中,文獻(xiàn)[7,8]中對(duì)行為函數(shù)的Lipschitz條件和文獻(xiàn)[8]中對(duì)行為函數(shù)的反函數(shù)的Lipschitz條件為兩個(gè)不等式所替代,而參數(shù)限制條件為兩個(gè)線性矩陣不等式所替代.因此得到了CGNN全局漸近穩(wěn)定性的新結(jié)果.

      2 模型及假設(shè)

      5 結(jié) 論

      本文首先利用同胚映射原理討論了具多時(shí)滯廣義CohenGrossberg神經(jīng)網(wǎng)絡(luò)平衡點(diǎn)的存在性和唯一性,繼而應(yīng)用平衡點(diǎn)的存在性結(jié)果、線性矩陣不等式和構(gòu)造的Lyapunov泛函研究了上述系統(tǒng)的全局漸近穩(wěn)定性,所得結(jié)果優(yōu)化了現(xiàn)有文獻(xiàn)中關(guān)于全局漸近穩(wěn)定性的判據(jù),而且所給判據(jù)是有效而實(shí)用的.

      參考文獻(xiàn)

      [1] S ARIK, Z ORMAN. Global stability analysis of CohenGrossberg neural networks with time varying delays [J]. Phys. Lett. A,2005,341(5-6):410-421.

      [2] B T CUI, W WU. Global exponential stability of CohenGrossberg neural networks with distributed delays [J]. Neurocomputing,2008,72(1-3):386-391.

      [3] Z ORMAN, S ARIK. New results for global stability of CohenGrossberg neural networks with multiple time delays [J]. Neurocomputing,2008,71(16-18):3053-3063.

      [4] W WU, B T CUI, X Y LOU. Some criteria for asymptotic stability of CohenGrossberg neural networks with time varying delays [J]. Neurocomputing,2007,70(4-6):1085-1088.

      [5] J FENG, S XU. New criteria on global robust stability of CohenGrossberg neural networks with time varying delays [J]. Neurocomputing,2008,72(1-3):445-457.endprint

      [6] Z S WANG, H G ZHANG, W YU. Robust stability criteria for interval CohenGrossberg neural networks with time varying delay [J]. Neurocomputing, 2009,72(4-6):1105-1110.

      [7] Z Q ZHANG, D M ZHOU. Global robust exponential stability for secondorder CohenGrossberg neural networks with multiple delays [J]. Neurocomputing, 2009,73(1-3):213-218.

      [8] H J JIANG, J D CAO. BAMtype CohenGrossberg neural networks with time delays [J]. Mathematical and Computer Modelling, 2008,47(1-2):92-103.

      [9] H Y ZHAO, L WANG. Hopf bifurcation in CohenGrossberg neural network with distributed delays [J]. Nonlinear Analysis: Real World Applications,2007,8(1):73-89.

      [10]X B NIE, J D CAO. Stability analysis for the generalized CohenGrossberg neural networks with inverse Lipschitz neuron activations [J]. Computer and Math Appli, 2009, 57(9):1522-1536.

      [11]M FORTI, A TESI. New conditions for global stability of neural networks with application to linear and quadratic programming problems [J]. IEEE Trans Circuit System I, 1995,42(7):345-366.endprint

      [6] Z S WANG, H G ZHANG, W YU. Robust stability criteria for interval CohenGrossberg neural networks with time varying delay [J]. Neurocomputing, 2009,72(4-6):1105-1110.

      [7] Z Q ZHANG, D M ZHOU. Global robust exponential stability for secondorder CohenGrossberg neural networks with multiple delays [J]. Neurocomputing, 2009,73(1-3):213-218.

      [8] H J JIANG, J D CAO. BAMtype CohenGrossberg neural networks with time delays [J]. Mathematical and Computer Modelling, 2008,47(1-2):92-103.

      [9] H Y ZHAO, L WANG. Hopf bifurcation in CohenGrossberg neural network with distributed delays [J]. Nonlinear Analysis: Real World Applications,2007,8(1):73-89.

      [10]X B NIE, J D CAO. Stability analysis for the generalized CohenGrossberg neural networks with inverse Lipschitz neuron activations [J]. Computer and Math Appli, 2009, 57(9):1522-1536.

      [11]M FORTI, A TESI. New conditions for global stability of neural networks with application to linear and quadratic programming problems [J]. IEEE Trans Circuit System I, 1995,42(7):345-366.endprint

      [6] Z S WANG, H G ZHANG, W YU. Robust stability criteria for interval CohenGrossberg neural networks with time varying delay [J]. Neurocomputing, 2009,72(4-6):1105-1110.

      [7] Z Q ZHANG, D M ZHOU. Global robust exponential stability for secondorder CohenGrossberg neural networks with multiple delays [J]. Neurocomputing, 2009,73(1-3):213-218.

      [8] H J JIANG, J D CAO. BAMtype CohenGrossberg neural networks with time delays [J]. Mathematical and Computer Modelling, 2008,47(1-2):92-103.

      [9] H Y ZHAO, L WANG. Hopf bifurcation in CohenGrossberg neural network with distributed delays [J]. Nonlinear Analysis: Real World Applications,2007,8(1):73-89.

      [10]X B NIE, J D CAO. Stability analysis for the generalized CohenGrossberg neural networks with inverse Lipschitz neuron activations [J]. Computer and Math Appli, 2009, 57(9):1522-1536.

      [11]M FORTI, A TESI. New conditions for global stability of neural networks with application to linear and quadratic programming problems [J]. IEEE Trans Circuit System I, 1995,42(7):345-366.endprint

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