Stability Of Cellular Neural Networks

Stability is a key issue in applications of cellular neural networks (CNNs). In some applications where a CNN possesses many equilibrium points, it is essential that the network involved is completely stable in the sense that every trajectory converges to an equilibrium point.In other applications, it is desirable that the network possesses a unique and globally asymptotically stable equilibrium point for every external input. In this thesis, we mainly study the complete stability and the global asymptotic stability of cellular neural networks. We establish the equivalence between state stability and output stability and two new stability theorems for models of general neural networks with variable delays by using the Lyapunov function and functional methods, which are different frcffn standard those. We also establish a complete stability theorem similar to LaSalle's invariance principle for cellular neural networks. Then by using these stability theorems and the essence of piecewise linearity of the output function we obtain a number of global asymptotic stability and complete stability results, some of them are dependent on the delay parameter. We show that in the case where the network is composed of two cells the external inputs do no; influence the existence of stable equilibrium points in saturation regions and the complete stability of the network, moreover, the existence of stable equilibrium points implies complete stability. Furthermore, we give two examples of three-cells CNN to show that the existence of stable equilibrium points in saturation regions does not imply the complete stability of the network, and the complete stability depends also on the choice of external inputs. By using special stable matrices and matrix measure theory, we discuss exponential stability of cellular neural networks.