On the Stability and Stabilization of Nonlinear Non-Stationary Discrete Systems

Abstract

The study of many biological, economic and other processes leads to its modeling based on discrete equations. Currently, the need to develop a mathematical apparatus for the qualitative analysis of discrete equations is caused by the creation of digital control systems, processors and microprocessors as well as discrete methods of signal transmission in automatic control systems and other theoretical and technical problems. One of the important areas of the qualitative analysis of discrete equations is the stability problem. The main method for studying the stability of nonlinear differential, discrete, and other types of equations is the direct Lyapunov method. The aim of this paper is to develop the direct Lyapunov method in the study of the limiting behavior and asymptotic stability of nonlinear nonstationary discrete equations using the comparison principle. New theorems are proved that are applied in the stability problem of the well-known epidemiological model as well as in solving the stabilization problem of a nonlinear discrete controlled system. An example is shown illustrating a qualitative difference in the conditions of stabilization of non-stationary and stationary discrete systems.