Abstract
We consider a linear control system with an almost periodic matrix of coefficients. The control has a form of feedback and is linear in phase variables. It is assumed that the feedback coefficient is almost periodic and its frequency modulus, i.e. the smallest additive group of real numbers, including all Fourier exponents of this coefficient, is contained in the frequency module of the coefficient matrix.The following problem is formulated: choose such a control from an admissible set so that the closed system has almost periodic solutions, the frequency spectrum (a set of Fourier exponents) of which contains a predetermined subset, and the intersection of the solution frequency modules and the coefficient matrix is trivial. The problem is called the control problem of the spectrum of irregular oscillations (asynchronous spectrum) with a target set of frequencies.The aim of the work aws to obtain a necessary solvability condition for the control problem of the asynchronous spectrum of linear almost periodic systems with trivial averaging of coefficient matrix The estimate of the power of the asynchronous spectrum was found in the case of trivial averaging of the coefficient matrix.
Highlights
The aim of the work aws to obtain a necessary solvability condition for the control problem of the asynchronous spectrum of linear almost periodic systems with trivial averaging of coefficient matrix The estimate of the power of the asynchronous spectrum was found in the case of trivial averaging of the coefficient matrix
The problem is called the control problem of the spectrum of irregular oscillations with a target set of frequencies
Необходимым условием разрешимости задачи управления асинхронным спектром линейных почти периодических систем с нулевым средним значением матрицы коэффициентов являентся конечность спектра целевого множества
Summary
The aim of the work aws to obtain a necessary solvability condition for the control problem of the asynchronous spectrum of linear almost periodic systems with trivial averaging of coefficient matrix The estimate of the power of the asynchronous spectrum was found in the case of trivial averaging of the coefficient matrix. Требуется выбрать такую матрицу U(t) (коэффициент обратной связи), чтобы замкнутая система x = ( A(t) + BU (t)) x имела сильно нерегулярное почти периодическое решение x(t), спектр частот которого содержит заданное подмножество L. Предварительно заметим, что даже если свободная система x = A(t)x имеет сильно нерегулярные почти периодические решения, то задача управления спектром сильно нерегулярных колебаний остается содержательной в силу открытости вопроса о мощности спектра.
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