The stability of Cohen-Grossberg neural networks with time dependent delays

Background. The study is devoted to the analysis of stability in the sense Lyapunov Cohen-Grossberg neural networks with time-dependent delays. To do this, we study the stability of the steady-state solutions of systems of linear differential equations with coefficients depending on time and with delays, time dependent. The cases of continuous and impulsive 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, technology, and it is necessary to determine the boundaries of their possible application. 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 in terms of the coefficients of the systems of differential equations modeling the networks. Materials and methods. The study of stability is based on the application of the method of “freezing” time-dependent coefficients and the subsequent analysis of the stability of the solution of the system in the vicinity of the “freezing” point. In the analysis of systems of differential equations transformed in this way, the properties of logarithmic norms are used. Results. A method is proposed that makes it possible to obtain sufficient stability conditions for solutions of finite systems of linear differential equations with coefficients and with time-dependent delays. The algorithms are effective both in the case of continuous and impulsive disturbances. Conclusions. The proposed method can be used in the study non-stationary dynamical systems described by systems of ordinary linear differential equations with delays depending from time. The method can be used as the basis for studying the stability of Cohen-Grossberg neural networks with discontinuous coefficients and discontinuous activation functions. Similar results were previously obtained for Hopfield neural networks in the work: Boykov I., Roudnev V., Boykova A. Stability of solutions to systems of nonlinear differential equations with discontinuous right-hand sides. Applications to Hopfield artificial neural networks. Mathematics . 2022:1.