OPTIMAL STABILIZATION IN SYSTEMS OF DIFFERENCE EQUATIONS

Abstract

The authors analyze the issues of stabilization theory for difference systems using the second Lyapunov method, which is based on finding so-called Lyapunov functions and studying their behavior. The authors investigate the existence of optimal solutions and their continuous dependence on initial conditions, system parameters, and difference operators, which can be useful in designing and optimizing various systems. The authors use the second Lyapunov method to find optimal control and establish conditions for its existence. As a result of the study, the asymptotic stability of the system is established when applying optimal control. In the work systems with scalar control and diagonal matrix control, as well as with performance criterion of general type are considered. The results of the study include the development of new methods and algorithms for optimal system stabilization, which can be used for practical applications in various fields such as automatic control, robotics, electrical engineering, and others. The article provides basic theoretical information, experimental results, and established regularities in solving the problem. The main emphasis is on important discoveries, new solutions, and conclusions. It allows readers to familiarize themselves with the main research results and determine its relevance to the scientific field. The conditions for the existence of optimal stabilizing control are formulated, and a theorem on optimal stabilization in difference equations systems is proved. For difference systems with scalar control, the general form of the stabilizing control function is determined as a theorem. Similar problems were solved for the system with a diagonal optimization matrix in control and systems with a matrix in the general type performance criterion.