Eventos Anais de eventos
DINAME 2017
XVII International Symposium on Dynamic Problems of Mechanics
An application of Lurie problem in Hopfield neural networks
Submission Author:
Rafael Pinheiro , SP
Co-Authors:
Rafael Pinheiro, Diego Colón
Presenter: Rafael Pinheiro
doi://10.26678/ABCM.DINAME2017.DIN17-0206
Abstract
The goal of this work is to present an application of Lurie problem results in Hopfield Neural Networks, aiming its analysis in relation to stability and the existence of chaotic behavior. The Lurie problem appeared in 1947, because of an aircraft control problem, and currently it is considered solved for the case of a single control, e.g. Popov criterion. However, for multiple controls the Lurie problem is not solved completely, with respect to generalization of the problem and the extent of the results of cases of a single control for multiple controls. The Lurie problem basically is to find necessary and sufficient conditions for the asymptotic stability global (Absolute Stability) to an equilibrium point of a system of ordinary differential equations with nonlinear functions restricted to the 1st and 3rd quadrants. Systems with these characteristics have been known in the literature as type Lurie systems. To solve problems related to this type of system, which dealt with the analysis of absolute stability, It was used techniques in the stability of Lyapunov theory, towards enabling the improvement of this theory, as well as achieving significant results for stability and control theory. As relevant to the nonlinear dynamics area, may be indicated that the study of absolute stability laid the foundation for the development of modern control theory; in particular, the development of a new area of control called Robust Control. The Lurie problem also has contributed in areas such as chaos and neural networks, and provides through its results parameters for engineers. The Hopfield network originated in 1984 due to J. J. Hopfield, it is a nonlinear network and there is interest to engineering or physics. The same proved to be initially useful to recovery patterns, however currently have shown to be relevant to the biological research area and aroused theoretical interest. In our case, more specifically, one may consider the Hopfield network as a particular case of type Lurie systems. We will use as methodology results derived from the study of absolute stability of type Lurie systems to non-linear network analysis. We may also make use of techniques using LMI's. Numerical simulations will be presented. Our research is in initial stage, which treats on the Lurie problem with multiple controls and their application in neural networks. So far, we find the relationship between the Hopfield network with Lurie problem, and so, we proceed with some applications. In this sense, we apply the results of Lurie problem with multiple controls to neural Hopfield networks.
Keywords
Lurie problem, Hopfield neural network
