Complete Stability Of One Dimensional Nonsymmetric CNN

In this paper we face the problem of stability for monodimensional nonsym-metric Cellular Neural Networks(CNN).It is well known that the absence of periodic or chaotic behavior,which guarantees complete stability,is a requirement for many applications.We call that a CNN is completely stable, if any initial condition(except at most for a zero measure set) bring the system to an equilibrium point.As the dynamic behavior has high sensitivity to modifications of the template parameters, a general study of CNN stability is extremely difficult.Up till now,although the stability of 1-D symmetric CNN (A = (α, a, α)) and antisymmetric CNN(A = (α, a, -α)) have approximately been resolved , only a few sufficient conditions ensuring complete stability of 1-D CNN with nonsymmetric template(A = (α, a, β), |α| |β|) were given.The complete stability of two classes of special nonsymmetric CNN will be studied in this paper.At first,we discuss the CNN's stability when the parameter of the output function r 0, and the coefficients α = 0, β 0.Then,a sufficient condition is presented to ensure complete stability for another CNN system where r = 0, and α 0, β 0.There are diffirent methods to prove their complete stability of the CNN presented above. As to the former, we mainly use the definition of stability to prove it directly,first,analysing the eigenvalues of coefficient matrix of the stationary solutions in each subregion,then,affirming the existence and stability of the equilibrium point.For the latter,the complete stability is guarantee! in terms of constructing Lyapunov functions.