No 6 (2025)

The application of the Gershgorin circle theorem and some of its derivatives to estimating matrix eigenvalues is considered. The obtained results are developed to design a localization region for matrix eigenvalues with interval‑indefinite constant and non‑stationary elements. The concept of e‑circles is introduced to obtain more accurate estimates of these regions than when using Gershgorin circles. The obtained results are applied to the stability analysis of network systems, where it is shown that the proposed methods allows one to analyze a network with a much larger number of agents than when using methods for solving linear matrix inequalities in CVX and Yalmip/SeDuMi, as well as the eig (for calculating matrix eigenvalues) and lyap (for solving the Lyapunov equation) algorithms in MatLab. It is shown that if the developed methods are applied not to the system itself, but to the result obtained using the Lyapunov function method, then it is possible to study systems with matrices without diagonal dominance. This allowed us to consider the modification of the Demidovich condition for systems with non‑stationary parameters and the design of the control law for non‑stationary systems with matrices without diagonal dominance. All the obtained results are illustrated by numerical modeling.

This paper considers a conservative system admitting a family of single-frequency oscillations with a domain Ω. For the original system, an autonomous controlled (ε-corrected) system with a small gain is introduced; a given oscillation from the domain Ω is stabilized by constructing a cycle that attracts all trajectories from this domain together with its ε-neighborhood. A universal adaptive control law, acting as a nonlinear force linear in velocity, is designed to track the current value of potential energy during motion. The cycle is constructed for any system oscillation. As a result, a new class of autonomous controlled systems is obtained based on the conservative system, and the operating modes of this class are stabilized (in the large) cycles with any desired energy. Examples are provided.