Lyapunov stability in Cohen-Grossberg neural networks

This thesis aims to deeply examine the stability properties of continuous Cohen-Grossberg artificial neural networks. The focus of the study is to develop a new and efficient energy function using Lyapunov techniques. This energy function will be used to determine critical properties of the neural network, such as the existence of equilibrium points, uniqueness, and global asymptotic stability. Another important aspect of the study is the similarity of artificial neural networks to electrical circuits. This conceptual similarity makes the network resemble a circuit of resistors and capacitors. This analogy is compatible with the artificial neural network model previously developed by Hopfield. In fact, the Hopfield model is considered a special case of the Cohen-Grossberg artificial neural network. Lyapunov's direct method is the main analytical tool of this study. Using this method, a new energy function will be defined over the created RC electrical circuit. This energy function will be a critical tool to evaluate the stability properties of the network. In the study, some basic assumptions will be made for the stability of the Cohen-Grossberg artificial neural network. These assumptions were determined in accordance with the energy exchange theories of the Russian mathematician Lyapunov. Lyapunov analyzed the behavior and stability conditions of systems based on energy change. In the process of defining the energy function, resistor and capacitor circuit elements, which are the physical equivalent of the neural network, will be taken into account. These circuit elements will be used to create the mathematical structure of the energy function. With the help of the energy function and selected parameters, a system of differential energy equations will be created that explains the stability properties of the network. This system of equations represents a complex set of differential equations that can be solved using circuit elements such as resistors and capacitors.