Stability of Cohen–Grossberg Neural Networks with Time-Dependent Delays.

The work is devoted to the analysis of Lyapunov stability of Cohen–Grossberg neural networks with time-dependent delays. For this, the stability of steady solutions of systems of linear differential equations with time-dependent coefficients and time-dependent delays is analyzed. The cases of continuous and pulsed perturbations are considered. The relevance of the study is due to two circumstances. Firstly, Cohen–Grossberg neural networks find numerous applications in various fields of mathematics, physics, and technology, and it is necessary to determine the limits of their possible application in solving each specific problem. Secondly, the currently known conditions for the stability of the Cohen–Grossberg neural networks are rather cumbersome. The article is devoted to finding the conditions for the stability of the Cohen–Grossberg neural networks, expressed via the coefficients of the systems of differential equations simulating the networks. The analysis of stability is based on the method of "freezing" time-dependent coefficients and the subsequent analysis of the stability of the solution in a vicinity of the freezing point. The analysis of systems of differential equations thus transformed uses the properties of logarithmic norms. A method is proposed making it possible to obtain sufficient stability conditions for solutions of finite systems of nonlinear differential equations with time-dependent coefficients and delays. The algorithms are efficient both in the case of continuous and pulsed perturbations. The method proposed can be used in the study of nonstationary dynamical systems described by systems of ordinary nonlinear differential equations with time-dependent delays. The method can be used as the basis for studying the stability of Cohen–Grossberg neural networks with discontinuous coefficients and discontinuous activation functions. [ABSTRACT FROM AUTHOR]