Contributed ArticleStability analysis of delayed cellular neural networks
Introduction
Recently, theoretical and applied studies of the cellular neural network (CNN) model have been a new focus point of studies worldwide (Chua and Yang, 1988aChua and Yang, 1988b). It is known that the CNN is formed by many units called cells, the structure of the CNN being similar to that found in cellular automata, namely, any cell in a cellular neural network is connected only to its neighbor cells. A cell contains linear and non-linear circuit elements, which typically are linear capacitors, linear resistors, linear and non-linear controlled sources and independent sources. The circuit diagram and connection pattern implementing the CNN can be referred to in Chua and Yang (1988a). The CNN can be applied in signal processing and can also be used to solve some image processing and pattern recognition problems (Chua and Yang, 1988b). However, it is necessary to solve some dynamic image processing and pattern recognition problems by using delayed cellular neural networks (DCNN) (Roska and Chua, 1992). Now, the DCNN can only be described by delayed differential equations (namely, functional differential equations), in fact, the CNN is described by ordinary differential equations. The study of the stability of DCNN and CNN is known to be an important problem in theory and thus has important significance in both theory and application. Some results on the stability of CNN have been obtained (Chua and Yang, 1988a, Roska and Chua, 1992, Liao, 1994), but only a few authors (Civalleri and Gilli, 1993, Lu and He, 1997, Cao, 1998) have considered the stability of DCNN. The purpose of this paper is to derive some new sufficient conditions for the global asymptotic stability of DCNN by using the Lyapunov functional method, some analysis techniques and by constructing different Lyapunov functionals (Cao and Wan, 1997), which are independent of delays, namely, the stability is, in fact, unconditioned global stability; in other words, the DCNN is globally asymptotically stable under our conditions for any delays. Civalleri and Gilli (1993) obtained only a stable sufficient condition dependent of delays, and Lu and He, 1997, Cao, 1998, studied some special conditions of the DCNN and obtained some unconditioned stable sufficient criteria. Our results extend and improve the results reported by Lu and He, 1997, Cao, 1998.
In this paper, we consider the following DCNN model described by differential equations with delaysin which n corresponds to the number of units in a neural network, xi(t) corresponds to the state vector of the ith unit at time t, fj(xj(t)) denotes the output of the jth unit at time t, aij, bij, Ii, ci are constant, aij denotes the strength of the jth unit on the ith unit at time t, bij denotes the strength of the jth unit on the ith unit at time t−rj, Ii denotes the external bias on the ith unit, rj corresponds to the transmission delay along the axon of the jth unit and is not negative constant, and ci represents the rate with which the ith unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs. In the following, we assume that each of the relations between the output of the cell fi(i=1,2,…,n) and the state of the cell possess the following properties:
(H1) fi(i=1,2,…,n) is bounded on R;
(H2) there is a number μi>0 such that |fi(u)−fi(v)|≤μi|u−v| for any u,v∈R.
It is easy to find from (H2) that fi is a continuous function on R. In particular, if the relation between the output of the cell and the state of the cell is described by a piecewise-linear function fi(x)=(1/2)(|x+1|−|x−1|), then it is easy to see that the function fi clearly satisfies the hypotheses (H1) and (H2) above, and μi=1(i=1,2,…,n). The circuit implementation of Eq. (1)can be referred to in Chua and Yang, 1988a, Roska and Chua, 1992.
Section snippets
Stability analysis of the DCNN
Lemma 1. Assume that the output of the cell function fi(i=1,2,…,n) satisfies the hypotheses (H1) and (H2) above. Then there exists an equilibrium for the DCNN Eq. (1).
Proof. If denotes an equilibrium of the DCNN Eq. (1), then satisfies the non-linear algebraic system of equations
Let B=[(1/ci)(aij+bij)]n×n, I=((I1/c1),(I2/c2),…,(In/cn))T, . Then Eq. (2)can be written
Conclusion
In this paper, we did not assume the symmetry of the connection matrix (aij)n×n,(bij)n×n and we only consider the output of the cell (i.e. the non-linear properties of the cell) using hypotheses (H1) and (H2) above and do not require them to be differentiable or strictly monotonously increasing. For this reason, the sufficient conditions established in the theorem above have a wider adaptive range, and these conditions can be applied to the design of unconditioned globally stable delayed
Acknowledgements
The authors wish to thank the referee and editor for valuable suggestions and revisions of this paper. This work is supported by the Natural Science Foundation of Yunnan Province, China.
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