Stability and periodicity for impulsive neural networks with delays
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RESEARCH
Open Access
Stability and periodicity for impulsive neural networks with delays Yuanfu Shao1* and Qianhong Zhang2 * Correspondence: [email protected] 1 College of Science, Guilin University of Technology, Guilin Guangxi, 541004, P.R. China Full list of author information is available at the end of the article
Abstract By constructing a suitable Lyapunov functional and using some inequalities, we investigate the existence, uniqueness and global exponential stability of periodic solutions for a class of generalized cellular neural networks with impulses and time-varying delays. Finally, an illustrative example and simulations are given to show the effectiveness of the main results. Keywords: periodicity; stability; impulsive; delays
1 Introduction The dynamics of cellular neural networks has been deeply investigated due to its applicability in solving image processing, signal processing and pattern recognition problems [, ]. Recently, the study of the existence and exponential stability of periodic solutions for neural networks has received much attention and many known results have been obtained [–]. For example, authors [] considered the following neural networks with delays:
x˙ i (t) = –ai (t)xi (t) +
n
n bij (t)fj xj (t) + cij (t)fj xj t – τij (t)
j=
+
n n
j=
n n dijl (t)fj xj (t) fl xl (t) + eijl (t)fj t – σijl (t)
j= l=
× fl xl t – σijl (t) + Ii (t),
j= l=
()
when τij (t) = τij , σijl (t) = σijl are constants, a set of easily verifiable sufficient conditions guaranteeing the existence and globally exponential stability of one periodic solution was derived. However, in practice, impulsive effects are inevitably encountered in implementation of networks, which can also be found in information science, electronics, automatic control systems and so on (see [, , –, , , –]). Thus it is necessary to study the impulsive case of system (). Motivated by the discussion, in this paper, by employing some inequalities and constructing a suitable Lyapunov functional, we aim to investigate the existence and exponential stability of a periodic solution for a class of nonautonomous neural networks with ©2013 Shao and Zhang; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Shao and Zhang Advances in Difference Equations 2013, 2013:352 http://www.advancesindifferenceequations.com/content/2013/1/352
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impulses and time-varying delays as follows: x˙ i (t) = –ai (t)xi (t) +
n
n bij (t)fj xj (t) + cij (t)fj xj t – τij (t)
j=
+
n n
j=
n n dijl (t)fj xj (t) fl xl (t) + eijl (t)fj t – σijl (t)
j= l=
()
j= l=
× fl xl t – σijl (t) + Ii (t), xi (tk ) = eik xi tk– , t = tk ,
t = tk ,
where i, j, l = , , . . . , n, n correspond
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