Piezoceramic products are actively used in modern technical devices and appliances. Disk piezoelectric devices are widely used in elements of information systems: in wireless communication, elements of satellite communication, global positioning systems. Among such devices, microwave piezo motors on traveling waves of various types are distinguished, including a motor built on an electromechanical disk converter, which performs non-axisymmetric oscillations under the action of a harmonic electric load. In this study, we perform separation of the electrode coating of the plate with thin diametrical dielectric sections and introduce a time-varying electric potential difference of different amplitude or phase to individual parts of the plate. We study the problem of non-axisymmetric planar oscillations of a piezoceramic disk with thickness polarization and the problem of optimization of the electromechanical coupling coefficient (EMCO) value as a function of an infinite number of parameters in space $l_2$ of square summable sequences.

The main object of research of this work is infinite-dimensional functional-differential equations of neutral type in Banach spaces. The conditions for the existence of a mild solution to the initial problem and its extension to the boundary of the domain are established. The presence of a delay in the derivative leads to the appearance of singular terms. The research methods are related to the technique of strong semigroups and fractional powers of the operator. The existence proof is based on the representation of the initial problem in abstract operator form with the further use of Krasnoselsky's theorem on a fixed point. For this purpose, the original operator is represented as the sum of the compression operator and the compact operator. The obtained abstract result is applied to the functional-differential equation in partial derivatives of the reaction-diffusion type.

Processes that involve jump-like changes are observed in mechanics (the movement of a spring under an impact; clockwork), in radio engineering (pulse generation), in biology (heart function, cell division). Therefore, high-quality research of pulse systems is a relevant task in the modern theory of mathematical modeling. This paper considers the issue related to the existence of bounded solutions along the entire real axis (semi-axis) of the weakly nonlinear systems of differential equations with pulse perturbation at fixed time moments. A concept of the regular and weakly regular system of equations for the class of the weakly nonlinear pulse systems of differential equations has been introduced. Sufficient conditions for the existence of a bounded solution to the heterogeneous system of differential equations have been established for the case of poorly regularity of the corresponding homogeneous system of equations. The conditions for the existence of singleness of the bounded solution along the entire axis have been defined for the weakly nonlinear pulse systems. The results were applied to study bounded solutions to the systems with pulse action of a more general form. The established conditions make it possible to use the classical methods of differential equations to obtain statements about solvability and the continuous dependence of solutions on the parameters of a pulse system. It has been shown that classical qualitative methods for studying differential equations are mainly naturally transferred to dynamic systems with discontinuous trajectories. However, the presence of a pulse action gives rise to a series of new specific problems. The theory of systems with pulse influence has a wide range of applications. Such systems arise when studying pulsed automatic control systems, in the mathematical modeling of various mechanical, physical, biological, and other processes.

Optimal Control for Functional-Differential Equation of Neutral Type in BanachТ‘Т‘ Spaces

Mathematical Olympiad is an intellectual competition that gives talented students the opportunity to demonstrate their knowledge, logical thinking and ability to solve complex mathematical problems. Olympiads stimulate in-depth study of mathematics, expand knowledge and improve students' skills. The work presents solutions to the problems of the 2024 All-Ukrainian Mathematics Olympiad of Taras Shevchenko National University of Kyiv were considered.

The purpose of the work is to characterize the methodological foundations of the formation and development of mathematical and information-digital competences of students of general secondary education institutions as a tool for forming their abstract thinking for decision-making in conditions of uncertainty. It is based on the concept of the development of abstract thinking as a process of mental activity of an individual as a higher form of thinking, which allows to operate with concepts, symbols, abstract ideas and representations, aimed at solving complex problems and making decisions in the conditions of underdefined or overdefined systems. The effectiveness of the proposed pedagogical technology is ensured by a child-centered orientation to the system of educational tasks that determine the formation of a system of abstract concepts, understanding their essence, and the development of applied skills in operating these categories in applied aspects. The main advantage of this pedagogical technology is ensuring that students learn the basic methods of solving educational problems, the application of mathematical competence for the development of information and digital competences in the conditions of digitization of education. Such a combination is a key factor in increasing the effectiveness of their educational activities in modern conditions of youth potential development. The proposed concept is an effective tool for the formation of students' mathematical and information-digital competences in the process of studying mathematics in the context of combined learning in indoor secondary education institutions, including in distance mode using synchronous and asynchronous communication tools.