Open AccessSolvability criterion of the control problem of an asynchronous spectrum of linear almost periodic systems with the trivial averaging of the coefficient matrix
Author(s) -
А. К. Деменчук
Publication year - 2020
Publication title -
doklady nacionalʹnoj akademii nauk belarusi
Language(s) - English
Resource type - Journals
eISSN - 2524-2431
pISSN - 1561-8323
DOI - 10.29235/1561-8323-2019-63-6-654-661
Subject(s) - mathematics , coefficient matrix , matrix (chemical analysis) , fourier series , linear system , spectrum (functional analysis) , mathematical analysis , fourier transform , asynchronous communication , control theory (sociology) , eigenvalues and eigenvectors , control (management) , computer science , physics , materials science , quantum mechanics , composite material , computer network , artificial intelligence
A linear control system with an almost periodic matrix of coefficients and the control in the form of feedback linear in phase variables is considered. It is assumed that the feedback coefficient is almost periodic and its frequency module, i. e. the smallest additive group of real numbers, including all the Fourier exponents of this coefficient, is contained in the frequency module of the coefficient matrix. The system under consideration is studied in the case of a zero average value of the matrix of coefficients. For the described class of systems, the control problem of the spectrum of irregular oscillations (asynchronous spectrum) with a target set of frequencies is solved. This task is as follows: to construct such a control from an admissible set so that the system closed by this control has almost periodic solutions, the set of Fourier exponents (frequency spectrum) that are contained in a predetermined subset; the intersection of the solution frequency modules and the coefficient matrix is trivial. The necessary and sufficient conditions for solvability of the control problem of the asynchronous spectrum are obtained.