Open AccessASYMPTOTIC SOLUTION OF THE PROBLEM OF OPTIMAL CONTROL OF SYSTEMS WITH SLOWLY VARIABLE PARAMETERS
Author(s) -
Rashevs’kyi Mykola
Publication year - 2020
Publication title -
naukovì pracì donecʹkogo nacìonalʹnogo tehnìčnogo unìversitetu. serìâ, problemi modelûvannâ ta avtomatizacìï proektuvannâ dìnamìčnih sistem
Language(s) - English
Resource type - Journals
ISSN - 2074-7888
DOI - 10.31474/2074-7888-2020-2-5-11
Subject(s) - mathematics , method of matched asymptotic expansions , variable (mathematics) , optimal control , differential equation , asymptotic analysis , matrix (chemical analysis) , operator (biology) , control variable , mathematical analysis , mathematical optimization , biochemistry , chemistry , materials science , statistics , repressor , transcription factor , composite material , gene
Models of non-stationary automatic control systems are differential equations with variable coefficients. Such equations do not integrate in quadratures in the general case. Asymptotic methods are methods of approximate integration of differential equations with variable coefficients. In the article the non-stationary automatic control system with slowly variable parameters is considered. To study this system it is necessary to construct an asymptotic representation of its solution. In the theory of asymptotic integration exist a problem to construction of the asymptotic solution of a system in the presence of a turning point. Special methods have been developed to construct a solution to such systems: Maslov’s canonical operator, the multiphase Kucherenko method, the method of W. Wasow. The purpose of the article is to construct an asymptotic solution of a linear system of differential equations with nonstabilitu spectrum of the main matrix. In this article the asymptotic representation of the solution of the optimal correction problem is constructed. The case of nonstability spectrum of the main matrix and the available of turning points are investigated. Application of the Pontryagin maximum principle to the problem leads to a system with slowly varying coefficients and an nonstable spectrum. Construction of a formal solution of the main system with turning points in the form of a single expression in some cases is possible. The system formed in the process of solving the problem of optimal correction does not allow the mentioned construction. A multiscale method was used to solve this system of equations. Asymptotic estimates for the constructed approximations are given. The studied problem has practical applications in technical and economic systems, in particular in the calculation of the correction of the orbits of artificial satellites. The nonnstability of the spectrum is the cause of the spike phenomenon. Further research may be aimed at finding a unified approach to solving such problems and to ascertain the physical meaning of the turning point in specific systems of automatic control. Keywords: automatic control system, asymptotic solutions, turning point, the problem of optimal correction.