D. S. Evtina, A. P. Zhabko

Saint Petersburg State University (Saint Petersburg, Russian Federation)

Abstract. This paper presents the algorithm for estimating solutions of differential- difference systems of neutral type with two incommensurate delays in the neutral part. It is worth mentioning an important assumption about the commutativity of matrices in the left-hand side of the system. The idea of the approach is to represent the system’s solutions in terms of initial functions and the fundamental matrix and then to construct an exponential estimate for this representation. At the first step, the system’s initial conditions are set. Next, the system is rewritten in an integral form and the delay operator is introduced. After recursive application of this operator to the right-hand side of obtained system, the system’s solutions are expressed via binomial coefficients, initial functions and the fundamental matrix. At the final step these expressions are used to make an exponential estimate of the solution. It is proved that the estimate of the fundamental matrix of the system also has an exponential form. In practice, the proposed method allows optimizing the control choice for neutral-type delay systems in sense of one of the crucial characteristics of the controlled systems, i.e. the overshoot value.

Key Words: differential equations, time-delay systems, neutral type delay, incommensurate delays

For citation: D. S. Evtina, A. P. Zhabko. Estimation of solutions of neutral type systems with two incommensurate delays. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 27:3(2025), 302–314. DOI: https://doi.org/10.15507/2079-6900.27.202503.302-314

Information about the authors:

Diana S. Evtina, Postgraduate Student, Department of Control Theory, Saint Petersburg State University (Universitetsky av., 35, Saint Petersburg 198504, Russia), ORCID: http://orcid.org/0009-0007-5417-606X, diana.evtina@mail.ru

Alexey P. Zhabko, D.Sc. (Phys. and Math.), Head of the Department of Control Theory, Saint Petersburg State University (Universitetsky av., 35, Saint Petersburg 198504, Russia), ORCID: https://orcid.org/0000-0002-6379-0682, a.zhabko@spbu.ru