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Registration Number

0218U001919, 0118U005313 , R & D reports

Title

Classification methods of theory of approximation, theory of boundary-value problems and their applications for the control of the complex mechanical systems.

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Head

Chaichenko Stanislav Olehovich,

Registration Date

21-12-2018

Organization

State Higher Educational Institution "DONBAS STATE PEDAGOGICAL UNIVERSITY

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The object of research is the extremal problems of the theory of approximation and the theory of convergence of mappings, matrix boundary-value problems for various systems of ordinary differential, functional-differential and differential-algebraic equations, and also problems of controllability and approximate controllability of mechanical systems in Hilbert spaces with finite-dimensional control. The purpose of the project is to study of the approximation properties of multipliers in modular sequence spaces, to obtain direct and inverse theorems of the approximation theory in Orlicz type spaces, to solve certain problems of convergence of mappings in metric spaces. The specific purpose of the project is to obtain of constructive conditions for the existence and to construct of effective algorithms for finding solutions of nonlinear matrix differential, functional-differential and differential-algebraic boundary-value problems, and constructive representation of the conditions of the approximate controllability of models of robotic systems whose evolution is given by infinite systems of differential equations with respect to Fourier coefficients of functions of elastic deformation. The subject of research is the best approximations and widths, the set of multipliers, the direct and inverse theorems of the theory of approximations, the conditions of convergence of mappings in metric spaces, iterative procedures for solutions of nonlinear integral and differential algebraic boundary value problems, the functions of control in the form of trigonometric polynomials, classes of mechanical systems with an unstable displacement and a system with incomplete dissipation in conditions of complete controllability. Methods of research are the methods of the theory of approximation and the theory of extremal problems, the apparatus of pseudo-inversion (by Moore-Penrose) of matrices and projections, the least squares method, the Lyapunov-Poincare methods, the construction of generalized Green's operators, the methods of nonlinear mechanics, and the methods of geometric control theory. In the first section there is to present the results of investigations of approximative characteristics of Orlicz type spaces. The exact values of the value of the best approximations, basic widths and Kolmogorov widths are found for some sets of images of multipliers in the Orlicz modular spaces, and the space of all multipliers between these spaces is described. As a result of the obtained results, the space of all multipliers acting between sequence spaces with variable exponent is described. In addition, the direct and inverse theorems of the approximation theory are obtained in the Orlicz type spaces in the terms of the corresponding modulus of continuous fractional order. In the second section, there is to obtain the theorems of convergence for one subclass of mappings with finite distortions that act between two metric spaces. Theorems of convergence and local behavior of reflections are the subject of research many well-known world mathematicians (Yu. Reshetnyak, O. Martio, M. Vuorinen, V. Gutlyanskii, V. Ryazanov and others). Note that on the basis of these theorems, the existence theorems can be obtained solutions of differential equations such as Beltrami type equations. At the moment these theorems are established for wide classes of mappings, but mainly only in the Euclidean space. This section is devoted of mappings of metric spaces. In the third section there is to presents the results of studies on the modification of the Newton-Kantorovich classical method for Banach spaces, which is a continuation of the studies of S. Campbell, V.F. Boyarintseva, A.M. Samoilenko, M.O. Perestyuk, V.P. Yakovets and O.A. Boychuk concerning the study of differential algebraic boundary-value problems. To find the solution of a nonlinear operator equation, an iterative scheme with quadratic convergence is constructed. It is shown that the modified version of the Newton-Kantorovich method can be used to find approximations to solutions of nonlinear integral and differential-algebraic boundary-value problems. In the fourth section there is to presents the results of studies of condition for the approximation of a gradient stream corresponding to the Lyapunov function, by the trajectories of systems of differential equations with limited displacement using control functions in the form of trigonometric polynomials. The coefficients of these trigonometric polynomials are also found in the terms of solutions of auxiliary algebraic equations on condition to execution of certain resonance ratios. The class of the considered differential equations contains mathematical models of mechanical systems with unstable displacement and systems with incomplete dissipation in conditions of complete controllability for which the parametrization of non-stationary feedback functions according to the state is obtained.

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Зуєв Олександр Леонідович
Нєсмєлова Ольга Володимирівна
Севостьянов Євген Олександрович
Чуйко Сергій Михайлович

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2020-04-02