Novel stability of cellular neural networks with interval time-varying delay

Abstract

In this paper, the asymptotic stability is investigated for a class of cellular neural networks with interval time-varying delay (that is, $0<h_1<d\left(t\right)<h_2$). By introducing a novel Lyapunov functional with the idea of partitioning the lower bound $h_1$ of the time-varying delay, a new criterion of asymptotic stability is derived in terms of a linear matrix inequality (LMI), which can be efficiently solved via standard numerical software. The criterion proves to be less conservative than most of the existing results, and the conservatism could be notably reduced by thinning the delay partitioning. Two examples are provided to demonstrate the less conservatism and effectiveness of the proposed stability conditions.

Introduction

Cellular neural networks (CNNs), which were introduced in Chua and Yang, 1988a, Chua and Yang, 1988b, have drawn increasing interest over the past few decades owing to their broad applications in a variety of areas, such as signal processing, pattern recognition, associative memory, and combinational optimization. Since time delay is frequently encountered and often causes instability and oscillations (Shi, Boukas, & Agarwal, 1999), the stability of neural networks (NNs) with time delay has received much attention in recent years. Various sufficient criteria for NNs stability have been proposed, either delay-independent or delay-dependent (Arik, 2002, Lu et al., 2003, Wang et al., 2006). Since delay-independent criteria are conservative, especially when the size of delay is small, much attention has been paid to the latter ones. Based on different assumptions and different approaches, a great number of stability criteria for delayed NNs have been proposed (see Chen and Rong (2004), Ensari and Arik (2005a), Wang, Ho, and Liu (2005) and Zhang, Shi, and Qiu (2001) and the references therein). Among them, many results have been concerned with constant delay. For example, in Gao, Lam, and Chen (2006), Singh (2004), Xu, Lam, Ho, and Zou (2004) and Zeng and Wang (2006), the authors have considered the CNNs with discrete and distributed constant delays, and obtained several sufficient conditions to ensure the existence and globally asymptotic stability of the equilibrium point for neural networks. In Wang, Liu, and Liu (2005), Xu et al., 2005a, Xu et al., 2005b, the globally exponential and robust stability problem has been investigated for CNNs with constant delays. Very recently, in He, Wu, and She (2006) and Zeng and Wang (2006), the authors have derived some criteria for CNNs with time-varying delays by the Lyapunov–Krasovskii approach.

Most of the existing results related to time-varying delay systems are based on the assumption $0<d\left(t\right)\le h_2$ (Ensari and Arik, 2005b, He et al., 2006, Mou et al., 2008, Xu et al., 2005a, Xu et al., 2005b, Zeng and Wang, 2006). However, in many practical systems, the typical delay may exist in an interval ($0<h_1\le d\left(t\right)\le h_2$), that is, the range of delay varies in an interval for which the lower bound is not restricted to 0. Typical examples of systems with interval time-varying delay are networked control systems (Gao et al., 2008, Yue et al., 2004). The aforementioned stability criteria for NNs with the assumption $0<d\left(t\right)\le h_2$, when applied to such cases, may be inevitably conservative due to their ignorance of the lower delay bound $h_1$. Therefore, it is of great significance to investigate the stability of systems with interval time-varying delay, which has driven the initial study on CNNs with interval time-varying delay (He, Liu, Rees, & Wu, 2007).

He et al. (2007) initiated the study on CNNs with interval time-varying delay, and presented some elegant stability conditions. In order to overcome the conservativeness, they paid careful attention to the time derivative calculation of the Lyapunov–Krasovskii functional (LKF), combined with the free-weighting matrix approach, which proves to be very effective. However, it is our observation that the results presented in He et al. (2007) could be further significantly improved if we employ the idea of delay partitioning (Mou et al., 2008), which motivates the present study. More specifically, we represent the time delay $d\left(t\right)$ as two parts: constant part $h_1$ and time-varying part $h\left(t\right)$, that is,

$$d\left(t\right)=h_1+h\left(t\right)\text{,}$$$$0\le h\left(t\right)\le h_2-h_1\text{.}$$

Then we introduce a novel Lyapunov–Krasovskii functional by applying the idea of delay partitioning to the constant part $h_1$. By utilizing the most updated techniques for achieving delay dependence, a new condition is proposed for the asymptotic stability of CNNs with time-varying delays, in the form of linear matrix inequality (LMI). Two examples are given to demonstrate the effectiveness and less conservatism of the proposed results over the existing ones in the literature.

The notation used throughout the paper is fairly standard. ${\mathbb{R}}^n$ denotes the $n$-dimensional Euclidean space and the notation $P>0$ ($\ge 0$) means that $P$ is real symmetric and positive definite (semi-definite). $I$ and 0 denote the identity matrix and zero matrix with compatible dimensions. In symmetric block matrices or complex matrix expressions, we use an asterisk $(\ast )$ to represent a term that is induced by symmetry. $\mathrm{diag}\{\dots \}$ stands for a block-diagonal matrix, and $\mathrm{sym}\left(A\right)$ is defined as $A+A^T$. Matrices are assumed to be compatible for algebraic operations if their dimensions are not explicitly stated.