一類非自治微分積分方程的全局指數穩定性

馬 超, 黎定仕

(西南交通大學數學學院,四川成都610031)

1 預備知識

微分積分方程穩定性的問題具有其特定的物理意義,許多學者對其進行了深入的探究[1-6].微分積分方程在多個科學領域中,如控制理論、生物、經濟、醫學等都會遇見,考慮其后效反應或者時滯狀態[7-8]已經成為了必要.特別地,人們常常用微分積分方程來描述具有遺傳性質的模型.而在這些領域中常見的時滯現象包括常數時滯和變量時滯[9-14],但是由于存在大量軸突大小和長度類似的路徑,微分積分方程常常會有空間上的外延性.于是,會有沿著這些路徑的傳導速度和傳播時滯的不同的現象產生.在這種情況下,信號的傳播不再是瞬間的,也不能用離散時滯來模擬,從而就出現了一種更為合適的描述,即連續的分布式時滯.

本文研究如下非自治微分積分方程

2 主要結果

3 例子

[1]Zhao H Y.Global asymptotic stability of Hopfield neural network involving distributed delays[J].Neural Networks,2004,17:47-53.

[2]Zhang Q,Wei X P,Xu J.Global exponential stability of Hop-field neural networks with continuously distributed delays[J].Phys Lett,2003,A315:431-436.

[3]Tian H J.The exponential asymptotic stability of singularly perturbed delay differential equations with a bounded lag[J].J Math Anal Appl,2002,270:143-149.

[4]龍述君.具有分布時滯的脈沖Cohen-Grossberg神經網絡的指數穩定性[J].四川師范大學學報:自然科學版,2009,32(1):68-71.

[5]莊劉,龍述君.一類中立型隨機泛函微分方程的穩定性分析[J].四川師范大學學報:自然科學版,2011,34(4):488-492.

[6]龍述君,張永新,向麗.具有混合時滯的隨機細胞神經網絡的穩定性分析[J].四川師范大學學報:自然科學版,2012,35(6):796-801.

[7]Hu L,Gao H J,Zheng W X.Novel stability of cellular neural networks with interval time-varying delay[J].Neural Networks,2008,21:1458-1463.

[8]Wang X H,Guo Q Y,Xu D Y.Exponentialp-stability of impulsive stochastic Cohen-Grossberg neural networks with mixed delays[J].Math Comput Simul,2009,79:1698-1710.

[9]Liao X F,Wong K W,Li C G.Global exponential stabilityfor a class of generalized neural networks with distributed delays[J].Nonlinear Anal:RWA,2004,5:527-547.

[10]Xu D Y,Yang Z C.Impulsive delay differential inequality and stability of neural networks[J].J Math Anal Appl,2005,305:107-120.

[11]Yang Z C,Xu D Y.Stability analsis of delay neural networks with impulsive effects[J].Ieee Transactions on Circuits and Systems-Ⅱ:Express Briefs,2005,52:517-521.

[12]Xu L G,Xu D Y.Exponential stability of nonlinear impulsive neutral integro-differential equations[J].Nonlinear Anal,2008,69:2910-2923.

[13]Sun Y H,Cao J D.p-th moment exponential stability of stochastic recurrent neural networks with time-varying delays[J].Nonlinear Anal:RWA,2007,8:1171-1185.

[14]Zhang H G,Wang G.New criteria of global exponential stability for a class of generalized neural networks with time-varying delays[J].Neurocomputing,2007,70:2486-2494.

[15]Long S J,Xu D Y,Zhu W.Global exponential stability of impulsive dynamical systems with distributed delays[J].Electronic J Quanlitative Theory of Differential Equations,2007,10:1-13.