MATHEMATICS

Stability and oscillations of synchronization systems with multidimensional nonlinearities

A. P. Elsakov^a, V. B. Smirnova^ab, A. V. Proskurnikov<sup>c

a</sup> Saint-Petersburg State University of Architecture and Civil Engineering, 2-ya Krasnoarmeiskaya ul., 4, Saint-Petersburg, 190005, Russia
^b Saint-Petersburg State University, Universitetskaya nab., 7–9, Saint-Petersburg, 199034, Russia
^c Politecnico di Torino, Corso Duca degli Abruzzi, 24, Turin, 10014, Italy

Abstract: This paper is devoted to the study of the asymptotic behavior of a class of control systems with multidimensional periodic nonlinearities and a countable set of equilibria. Such systems are known as synchronization or pendulum-like systems. Their stability is defined as the convergence of any solution to one of the equilibria. The dynamics of synchronization systems cannot be studied using classical methods designed for systems with a single equilibrium state. Therefore, within the framework of the classical methods of A.M. Lyapunov and V.M. Popov, special methods have been developed to generate conditions for the convergence of solutions and conditions for the absence of oscillations of a certain frequency. For systems with multidimensional nonlinearities, these conditions take the form of matrix frequency inequalities with variable parameters. This paper substantiates the optimal choice of variable parameters, which allows obtaining improved estimates of stability regions and regions of absence of high-frequency oscillations in the parameter space of specific systems.

Keywords: synchronization system, frequency-domain criteria, high-frequency oscillations, global asymptotic stability

UDC: 517.9

MSC: 93D10, 45M05, 45M15

Received: 01.09.2025
Accepted: 02.11.2025

DOI: 10.35634/2226-3594-2025-66-03

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