scholarly journals Necessary condition for solvability of the control problem of an asynchronous spectrum of linear almost periodic systems with trivial averaging of the coefficient matrix

2019 ◽  
Vol 55 (2) ◽  
pp. 176-181 ◽  
Author(s):  
A. K. Demenchuk
Keyword(s):  
Control Problem ◽  
Solvability Condition ◽  
Additive Group ◽  
Coefficient Matrix ◽  
Periodic Systems ◽  
Necessary Condition ◽  
Almost Periodic ◽  
Real Numbers ◽  
Frequency Module ◽  
Target Set

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.

scholarly journals A sufficient condition for the unsolvability of the control problem of the asynchronous spectrum of linear almost periodic systems with the diagonal averaging of the coefficient matrix

2020 ◽  
Vol 56 (4) ◽  
pp. 391-397
Author(s):  
A. K. Demenchuk
Keyword(s):  
Control Problem ◽  
Additive Group ◽  
Coefficient Matrix ◽  
Periodic Systems ◽  
Sufficient Condition ◽  
Almost Periodic ◽  
Real Numbers ◽  
Average Value ◽  
Target Set ◽  
Special Case

We consider a linear control system with an almost periodic matrix of the coefficients. The control has the form of feedback that is linear on the 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 the Fourier exponents of this coefficient, is contained in the frequency modulus of the coefficient matrix. The following problem is formulated: choose a control from an admissible set for which the system closed by this control has almost periodic solutions with the frequency spectrum (a set of Fourier exponents) containing a predetermined subset, and the intersection of the frequency modules of solution and the coefficient matrix is trivial. The problem is called as the control problem of the spectrum of irregular oscillations (asynchronous spectrum) with the target set of frequencies. At present, this problem has been studied only in a very special case, when the average value of the almost periodic coefficients matrix of the system is zero. In the case of nontrivial averaging, the question remains open. In the paper, a sufficient condition is obtained under which the control problem of the asynchronous spectrum of linear almost periodic systems with diagonal averaging of the coefficient matrix has no solution.

scholarly journals Solvability of the control problem of the asynchronous spectrum of linear almost periodic systems with a lower triangular representation of the averaging of coefficient matrix

2021 ◽  
Vol 65 (3) ◽  
pp. 263-268
Author(s):  
A. K. Demenchuk
Keyword(s):  
Control Problem ◽  
Sufficient Conditions ◽  
Additive Group ◽  
Coefficient Matrix ◽  
Almost Periodic ◽  
Average Value ◽  
The Matrix ◽  
Frequency Module ◽  
Necessary And Sufficient ◽  
And Control

A linear control system with an almost periodic matrix of coefficients and control in the form of the 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 triangular average value of the matrix of coefficients. For the described class of systems, the control problem of the asynchronous spectrum with a target set of frequencies is solved. This task is to construct such a control from an admissible set that the system closed by this control has almost periodic solutions, a set of the Fourier exponents of which contains a predetermined subset, and the intersection of the solution frequency modules and the coefficient matrix is trivial. The necessary and sufficient conditions for the solvability of this problem are obtained.

scholarly journals Solvability criterion of the control problem of an asynchronous spectrum of linear almost periodic systems with the trivial averaging of the coefficient matrix

2020 ◽  
Vol 63 (6) ◽  
pp. 654-661
Author(s):  
A. K. Demenchuk
Keyword(s):  
Control Problem ◽  
Sufficient Conditions ◽  
Additive Group ◽  
Coefficient Matrix ◽  
Almost Periodic ◽  
Average Value ◽  
The Matrix ◽  
Solvability Criterion ◽  
Frequency Module ◽  
Necessary And Sufficient

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.

scholarly journals On linear almost periodic systems with bounded solutions

1997 ◽  
Vol 55 (2) ◽  
pp. 177-184 ◽  
Author(s):  
V. I. Tkachenko
Keyword(s):  
Differential Equations ◽  
Periodic Solutions ◽  
Linear Differential Equations ◽  
Periodic Systems ◽  
Almost Periodic Solutions ◽  
Almost Periodic ◽  
Bounded Solutions ◽  
Adjoint Matrix ◽  
Frequency Module ◽  
Linear Differential

It is proved that in every neighbourhood of a system of linear differential equations with almost periodic skew-adjoint matrix with frequency module ℱ there exists a system with frequency module contained in the rational hull of ℱ possessing all almost periodic solutions.

scholarly journals An alternative characterization for matrix exponential distributions

2009 ◽  
Vol 41 (04) ◽  
pp. 1005-1022
Author(s):  
Mark Fackrell
Keyword(s):  
Continuous Function ◽  
Exponential Distribution ◽  
Matrix Exponential ◽  
Necessary Condition ◽  
Stieltjes Transform ◽  
Real Numbers ◽  
Exponential Distributions ◽  
Alternative Characterization

A necessary condition for a rational Laplace–Stieltjes transform to correspond to a matrix exponential distribution is that the pole of maximal real part is real and negative. Given a rational Laplace–Stieltjes transform with such a pole, we present a method to determine whether or not the numerator polynomial admits a transform that corresponds to a matrix exponential distribution. The method relies on the minimization of a continuous function of one variable over the nonnegative real numbers. Using this approach, we give an alternative characterization for all matrix exponential distributions of order three.

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