Elsevier

Applied Mathematics and Computation

Volume 229, 25 February 2014, Pages 457-466
Applied Mathematics and Computation

Asymptotic stability of cellular neural networks with multiple proportional delays

https://doi.org/10.1016/j.amc.2013.12.061Get rights and content

Abstract

Proportional delay is a kind of unbounded time-varying delay, which is different from unbounded distributed delays. In this paper, asymptotic stability of the equilibrium point of cellular neural networks with multiple proportional delays is presented. Sufficient conditions for delay-dependent global asymptotic stability and delay-independent uniform asymptotic stability of the system are obtained by employing matrix theory and constructing Lyapunov functional. Two examples are given to illustrate the effectiveness of the obtained results and less conservative than previously existing results.

Introduction

Cellular neural networks (CNNs) [1] have been widely investigated owing to their widespread applications in image processing, pattern recognition, optimization and associative memories. These practical application of CNNs depends on the existence and stability of the equilibrium point of the CNNs. Moreover, time delay is inevitable due to the finite switching speed of information processing and the inherent communication time of neurons, and its existence may cause the instability of the system. Therefore, a number of stability criteria of CNNs with delays have been proposed [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20]. At present, stability results of CNNs with delays studied in [4], [6], [11], [13], [16], [18] are mainly based on such approaches as M-matrix, algebraic inequalities, and so on. As pointed out in [14], the characteristic of those results, which means to take absolute value operation on the interconnection matrix, leads to the ignorance of neuron’s inhibitory and excitatory effects on neural networks. In recent years, linear matrix inequality (LMI) technique has been used to deal with the stability problem for neural networks [2], [3], [5], [7], [8], [9], [10], [12], [14], [15], [17], [19], [20]. The feature of LMI-based results is that it can consider the neuron’s inhibitory and excitatory effects on neural networks. However, few stability results have been obtained for CNNs with proportional delays on basis of LMI, which is important, as did for neural networks model studied in [32], [33]. The exponential stability of CNNs with multi-pantograph delays was studied by nonlinear measure in [32]. In [33], by employing matrix theory and Lyapunov functional, global exponential stability of a class of CNNs with multi-proportional delays was investigated and delay-dependent sufficient conditions were obtained. The proportional delay system as an important mathematical model often rises in some fields such as physics, biology systems and control theory and it has attracted many scholars’ interest [25], [26], [27], [28], [29], [30], [31].

As a neural network usually has a spatial nature for the presence of an amount of parallel pathways of a variety of axon sizes and lengths, it is desired to model by introducing continuously proportional delay over a certain duration of time. The proportional delay function is that τ(t)=(1-q)t+ as q1,t+ (0<q1), so it is time-varying, unbounded, and monotonically increasing. Therefore, the network’s running time may be controlled according to the network allowed delays. At the present time, besides the distributed delays, the delay function τ(t) is usually required to be bounded in the stability discussion of neural networks with delays, such as, these results in [9], [12], [19], [20], [21], [22], [23], [24], are required that the delay function τ(t) satisfies 0τ(t)τ and other conditions. It is seldom that the delay function τ(t) is unbounded, i.e. τ(t)+(t+). Compared with the distributed delay [6], [7], [8], [10], [13], whose delay kernel functions satisfy some conditions such that the distributed delay is easier to be handled [8], [10], [13] in the use inequality, but the proportional delay is not easy to be controlled. Thus, it is important to study stability of neural networks with proportional delays in theory and practice.

Motivated by the discussion above, the asymptotic stability of CNNs with multiple proportional delays is further discussed in this paper, inspired by Liao et al. [4], Zhang and Wang [14] and Zhang and Zhou [32]. This paper is organized as follows. Model description and preliminaries are given in Section 2. Two delay-dependent and delay-independent sufficient conditions are given in Section 3 to ascertain the asymptotic stability of the CNNs with multiple proportional delays, which are easy to be verified. Numerical examples and their simulation are presented in Section 4 to illustrate the effectiveness and less conservative of obtained results. Finally, conclusions are given in Section 5.

Section snippets

Model description and preliminaries

Consider the following CNNs with multiple proportional delaysẋi(t)=-dixi(t)+j=1naijfj(xj(t))+j=1nbijfj(xj(qijt))+Ii,xi(s)=xi0,s[q,1],i=1,2,nfor t1, where n is the number of neurons; di>0 is a constant; xi(t) is the state variable; aij and bij are constants which denote the strengths of connectivity between the cells j and i at time t and qijt, respectively; fi(·) denotes a nonlinear activation function; Ii denotes the constant external inputs; qij,i,j=1,2,,n are proportional delay

Main results

Theorem 3.1

The origin of system (2.3) is globally asymptotically stable, if there exist positive diagonal matrices M=diag(m1,m2,,mn),Ni=diag(ni1,ni2,,nin) and a constant β>0, such that the following inequality holds:MA+ATM-2MDL-1+i=1n(βNiQi-1+β-1MWiNi-1WiTM)<0,where D=diag(d1,d2,,dn),A=(aij)n×n,L=diag(l1,l2,,ln),Wi is an n×n square matrix, whose ith row is composed of (bi1,bi2,,bin) and other rows are all zeros, i,j=1,2,,n and Qi-1=diag(qi1-1,qi2-1,,qin-1),i=1,2,,n.

Proof

Consider the following Lyapunov

Numerical examples

Example 4.1

Consider the neural network model (2.1), whereD=8006,A=-211-1,B=-0.5-1-0.3-0.6,q=0.50.80.80.5,I=00,the activation function is described by f(xi)=0.5(|xi+1|-|xi-1|)(i=1,2) with li=1,i=1,2. Clearly, f(xi),i=1,2 satisfy condition (2.2) above.

By some simple calculations, we obtain Q1=diag(0.5,0.8),Q2=diag(0.8,0.5), and L-1=diag(1,1). AndW1=-0.5-100,W2=00-0.3-0.6,Q1-1=diag(2,1.25),Q2-1=(1.25,2),

Take M=diag(2,3),N1=diag(3,4),N2=diag(1,3) and β=1. Applying Theorem 3.1, we have-MA+ATM-2MDL-1+i=12(βNiQi

Conclusions

New sufficient conditions are derived for asymptotic stability of equilibrium point for cellular neural networks with multiple proportional delays, which is different from the existing ones and has wider application fields. It can be shown that the derived criteria are less conservative than previously existing results through the numerical example. The obtained results can be expressed in the form of linear matrix inequality and be easy to be verified by utilizing Matlab LMI toolbox. These

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      In [32], the global asymptotic stability of equilibria in autonomous systems of differential equations are analyzed. In [33], the asymptotic stability of the equilibrium point of cellular neural networks with multiple proportional delays is presented. In order to improve the convergence rate of closed-loop system, the concept of fixed-time stabilization (FTS) [34,35] is introduced in this paper.

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    This work is supported by the National Science Foundation of China (Nos. 61374009, 61272514, 61003287, 61170272, 61121061, 61161140320), Tianjin Municipal Education Commission (No. 20100813), NCET (No. NCET-13-0681), the Specialized Research Fund for the Doctoral Program of Higher Education (No. 20100005120002), the Fok Ying Tong Education Foundation (No. 131067) and the Fundamental Research Funds for the Central Universities (No. BUPT2012RC0221).

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